Modeling the Earth-Moon Distance for Different Theories of Gravitation

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1 Wellesley College Wellesley College Digital Scholarship and Archive Honors Thesis Collection 2017 Modeling the Earth-Moon Distance for Different Theories of Gravitation Sanaea Rose Follow this and additional works at: Recommended Citation Rose, Sanaea, "Modeling the Earth-Moon Distance for Different Theories of Gravitation" (2017). Honors Thesis Collection This Dissertation/Thesis is brought to you for free and open access by Wellesley College Digital Scholarship and Archive. It has been accepted for inclusion in Honors Thesis Collection by an authorized administrator of Wellesley College Digital Scholarship and Archive. For more information, please contact

2 Modeling the Earth-Moon Distance for Different Theories of Gravitation by Sanaea Cooper Rose Submitted to the Department of Physics; Department of Astronomy in partial fulfillment of the requirements for the degree of Bachelor of Arts in Astrophysics at WELLESLEY COLLEGE May 2017 c Wellesley College All rights reserved. Author Department of Physics; Department of Astronomy May 2017 Certified by James Battat Assistant Professor Thesis Supervisor Accepted by Richard French; Glenn Stark Chair, Department of Astronomy; Chair, Department of Physics

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4 Modeling the Earth-Moon Distance for Different Theories of Gravitation by Sanaea Cooper Rose Submitted to the Department of Physics; Department of Astronomy on May 2017, in partial fulfillment of the requirements for the degree of Bachelor of Arts in Astrophysics Abstract Lunar Laser Ranging (LLR) measures the transit time of a laser pulse from an observatory on Earth to corner cube retroreflectors on the Moon, installed by Apollo astronauts and Russian rovers. A transit time measurement with 7 ps precision gives the Earth-Moon distance with millimeter precision. Successive measurements indicate the evolution of this distance and can constrain theories of gravitation. These theories include General Relativity (GR), which remains incompatible with Quantum Mechanics. This project presents simulations of the Earth-Moon-Sun system in Python that focus solely on the physics of gravitation and provide a means of comparing the evolution of the Earth-Moon distance for different gravitational frameworks. The comparisons thus far have focused on violations of the Strong Equivalence Principle (SEP) using a quasi-newtonian formulation of gravitation. The comparisons between the Newtonian and quasi-newtonian frameworks take two approaches. The first involves subtracting simulated Newtonian Earth-Moon distances from simulated SEP violation Earth-Moon distances and analyzing the results using a Fourier transform, which identifies periodic signals. The Fourier spectrum revealed an SEP violation signal as well as three other signals related to the Sun s perturbation of the Earth-Moon system and the eccentricity of the lunar orbit. However, this method is not ideal because the rate of precession of the lunar orbit depends on the degree of SEP violation, or lack thereof. The more optimal, second approach consists of fitting SEP violation simulated data with a Newtonian model. The post-fit residuals should be dominated by the SEP violation signal. The results were varied: the SEP violation signal dominated some of the post-fit residuals, while in some fits with different initial conditions, the other periodic signals persisted. The reasons for this inconsistency remain unclear. Because of correlations between parameters, the model was able to absorb some of the SEP violation signal by adjusting parameters like the initial velocities and positions of the objects. However, all fits failed to fully mask the SEP violation, indicating that LLR is sensitive to an SEP violation. Future work may focus on optimizing the fit and increasing its accuracy, and on undertaking similar comparisons for GR and alternative theories. 3

5 Thesis Supervisor: James Battat Title: Assistant Professor 4

6 Acknowledgments My thanks to Professor James Battat for his constant support, lucid explanations, and patience, which made physics a pleasure to learn and a challenge worth pursuing. I am also grateful to Professor Richard French for his guidance when implementing the Fourth Order Runge-Kutta method and the thesis committee members Professor Richard French, Professor Glenn Stark, Professor Jonathan Tannenhauser, and Professor Eve Zimmerman for their suggestions and support. The support of my parents chocolate, s with cat cartoons, FaceTime calls, and boundless affection has carried me through the late nights of wrestling with LaTex and Python. 5

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8 Contents 1 Introduction 17 2 Underlying Physics The Equivalence Principle and Quasi-Newtonian Acceleration The Nordtvedt Effect General Relativistic Acceleration Simulation Characteristics Numerical Method The Time Step Initial Conditions and Assumptions Precision Simulation Results Comparing Newtonian and quasi-newtonian Simulations The SEP Violation Signal The Amplitude of the Envelope and its Relation to η General Relativistic Acceleration Fitting the Simulated Data Perturbing the Parameters Fit Method Simple Tests of the Fit 65 7

9 6.1 Fitting a Single Parameter Fitting Multiple Parameters Fitting All Parameters When η O η Ci Fitting an SEP Violation with a Newtonian Model Fitting Newtonian Observations with a Full SEP Violation Model Implications Conclusion 79 A Derivations 83 A.1 Gravitational Self Energy of a Uniform Sphere A.2 Nordtvedt Effect B Code 89 B.1 Differencing Simulations and Taking a Fourier Transform B.2 Functions for Simulating Earth-Moon Distances B.3 Initial Conditions B.4 Fit Code C Duration and the Fourier Spectrum 101 8

10 List of Figures 1-1 The RMS of the residuals between observations and the range model and binned by year quantify the historical precision of LLR measurements. Apollo, which began taking data in 2006, produces range measurements with millimeter level precision. Figure courtesy of Professor James Battat The figure on the left shows the temporal spread of measurements in the absence of lunar libration convolved with the trapezoidal signature of lunar libration, which can be modeled. The figure on the right shows the spread of photon returns from a local corner cube reflector, with which the trapezoidal signature is convolved. Figure taken from reference [8] The system s energy appears to be conserved for four-year long simulations using the Euler method (left) and Fourth Order Runge-Kutta method (right). Both exhibit fluctuations in the energy, but neither has a net gain or loss of energy over the course of the simulation. The fluctuations are small for both cases. For the Euler method, the energy fluctuations are a part in 10 5 change in the total energy. The Fourth Order Runge-Kutta method is more optimal: the energy fluctuations are a part in 10 9 change in the total energy

11 3-2 The figures above plot the total energy of the system versus time for time steps of 2000 seconds (upper left), 1000 seconds (upper right), 500 seconds (lower left), and 100 seconds (lower right). The total energy corresponding to the 2000 second time step exhibits a noticeable net loss in energy over the four-year long simulation. The energy fluctuates slightly in each case. In the order of decreasing time step, the fluctuations E are 10 25, 10 24, 10 23, and Joules The system s energy versus time is plotted for four-year Newtonian and quasi-newtonian (η = 1) simulations using time steps of 1000, 500, and 100 seconds. The energy fluctuations for the Newtonian case are always Joules less than the energy fluctuations of the full SEP violation case even as the size of energy fluctuations decreases for both models Not to scale, the cartoon depicts the initial conditions of the simulation, which places the Sun, Earth, and Moon along the x-axis in the center of mass frame. Arrows denote initial velocities The velocities and positions of the Earth, Moon, and Sun are adjusted so that the center of mass remains stationary at the origin. Small numerical errors give the center of mass a non-zero position and velocity, which then changes slightly during the simulation. The coordinates for the center of mass position and velocity versus time are plotted above. While non-zero, both the velocity and position coordinates remain negligible, on the order of m/s and 10 8 meters, respectively Two three-year long simulations, one using Python s np.float64 option and the other, np.float128, were differenced to determine the effect of precision on the computed Earth-Moon distances. The differences between the two simulations were minimal. The distances never differed by more than four centimeters

12 4-1 The Earth-Moon distance for a two-year long simulation with a time step of 500 seconds exhibits a variety of periodic behavior that depends on the the positions of all three bodies with respect to each other. The average Earth-Moon distance is about kilometers, and the distance varies from about to kilometers The first plot shows the Earth-Moon distance over the four-year long simulations with η = 0 and η = 1. The lower plot consists of the differences between those two simulations, r(η = 1) r(η = 0). The differences have a wave-packet form that begins to disintegrate towards the third year r(η = 1) r(η = 0) for seven-year long simulations exhibits an almost identical form to that in Figure 4-2, for four-year long simulations. However, as the simulations progress, the wave packet continues to dissociate due to the slight differences in the accelerations, which, for example, impact the precession of the lunar orbit The Fourier spectrum of r(η = 1) r(η = 0) for four-year long simulations exhibits four prominent signals with periods corresponding to the the Nordtvedt effect (29.53 days), the Sun s perturbation of the Moon s orbit (31.8 days and days), and the elliptical shape of the lunar orbit (27.55 days). The periods of the signals in the spectrum are slightly less than the accepted values. The SEP violation signal has an amplitude of approximately 11 meters, slightly less than the 13 meter prediction Increasing the duration of the simulation from four to seven years increases the resolution of the Fourier Spectrum. The signals also increased in amplitude, with the SEP violation signal having an amplitude of 12.5 meters, closer to the predicted 13 meters

13 4-6 This figure presents the results from differencing two-year long SEP violation and Newtonian simulations. The resolution of the Fourier spectrum suffices to distinguish the SEP violation signal and the signal associated with the eccentricity of the lunar orbit. The amplitude (12.9 meters) and period (29.2 days) of the Nordtvedt Effect signal best match the expectations of days and 12.9 meters The amplitude of the envelope appears to be proportional to η. The figure above shows both r(η = 1) r(η = 0) and r(η = 0.5) r(η = 0) versus time. The latter has half the amplitude of the former, which the Fourier spectrum further emphasizes. The signals in the Fourier spectrum for the η = 0.5 case have half the amplitude of those for the η = 1 case The difference between six-year long GR and Newtonian simulations has the same envelope form as the other results in this chapter. The amplitude of the envelope and the signals in the Fourier spectrum seem less credible. The amplitudes most likely indicate an error in the implementation of the simulation The plots above show r versus time when a parameter is changed by p j, given in Table 5.1. r = r(p j = p j0 + p j ) r(p j = p j0 ). p j was adjusted to produce a maximum r of a meter over the course of the two-year simulation. The masses are in units of kilograms, velocities in meters per second, and position coordinates in meters The finite differences in the Earth-Moon distance computed at η = 1 and η = 0 with η = 0.05 are nearly indistinguishable for the first few years of the five-year long simulation. The differences become noticeable after three years

14 7-1 The SEP violation signal dominates the post-fit residuals of Fit A, which has a duration of two years. With a 10 meter signal at the synodic period, the Fourier spectrum indicates that in the absence of a non-zero η, the post-fit parameter values cannot mask an SEP violation Using a duration of three years, Fit B has post-fit residuals with an envelope shape. Both the residuals and the Fourier spectrum, with four signals, resemble the results from Chapter 4, in which Newtonian and full SEP violation simulations were differenced With a time step of 500 seconds, Fit C s results strongly resemble the results of Fit A, which had a time step of 1000 seconds With a duration of two years and a time step of 100 seconds, Fit D produced larger residuals which lacked both a shape dominated by the Nordtvedt Effect and a series of wave envelopes. The Fourier spectrum includes the SEP violation signal of 10 meters at the synodic period. However, it also includes two strong signals the strongest of the fits presented in this chapter around 14 and 27 days With a duration of three years and a time step of 1000 seconds, the most successful fit involved fitting Newtonian data with a full SEP violation model for a duration of three years. The only prominent signal in the Fourier spectrum of the residuals is the SEP violation signal Fitting Newtonian data with a full SEP violation model was less successful for a duration of two years than it was for a duration of three years. The Fourier spectrum of the post-fit residuals exhibit strong signals at periods other than the synodic period

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16 List of Tables 5.1 The table presents the nominal values for all the parameters after the center of mass has been fixed to the origin. The table also includes the fractional, when applicable, and absolute change in the parameters to produce a maximum r of one meter over a two-year long simulation. The masses are in units of kilograms, velocities in meters per second, and position coordinates in meters These four tests had the same observed and pre-fit computed parameter values for all parameters but η. The table lists post-fit computed values. The pre-fit, post-fit, and observed eta are each specified. The masses are in units of kilograms, velocities in meters per second, and position coordinates in meters The post-fit computed parameter values for Fit A and D differ from the pre-fit parameter values to compensate for the SEP violation. The parameter η for the model was fixed at 0. The masses are in units of kilograms, velocities in meters per second, and position coordinates in meters

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18 Chapter 1 Introduction Galileo Galilei famously challenged the misconception that objects of different weights fall at different rates by dropping objects from the Tower of Pisa. He asserted that scientific thought be grounded in empirical observations and privileged the happenings of the physical world over abstract thought. It is somewhat ironic, then, that his student Vincenzo Viviani did not extend the same level of commitment to fact, that is, documenting actual occurences, to his biography of Galileo. While Galileo did conduct experiments with pendulums, other than Viviani s description, no evidence exists that Galileo dropped objects from the Tower of Pisa. 1 Theatrics aside, this idea that objects interact with a gravitational field irrespective of their masses and compositions, today called the Weak Equivalence Principle (WEP), remains a fundamental assumption of modern formulations of gravitation. Less than a century after Galileo, Isaac Newton incorporated this principle into his theory of gravitation. He described gravity as a long-range force, one that acted over a distance and did not require contact between objects. The strength of the force is proportional to the masses involved and inversely proportional to the square of the distance between them. While Newtonian gravity reliably predicts the interactions between objects in many cases, a comprehensive theory of gravitation requires a far more complex for- 1 Segre, Michael. Viviani s Life of Galileo. Isis, vol. 80, no. 2, 1989, pp. 206?231., 17

19 mulation. Albert Einstein provided such a theory, General Relativity (GR), for which Newtonian gravity often functions as a convenient approximation. GR operates on the principle that mass curves spacetime. A common analogy compares this condition to the way that the weight of a bowling ball deforms the surface of a mattress on which it rests. A small nearby mass, like a marble, follows the curvature of the mattress and rolls into the indentation made by the bowling ball. Einstein reformulated the gravitational force felt by the other mass as its tendency to follow the curvature of spacetime. GR has a number of observable effects in the Solar System. For example, geodetic precession occurs if there is a rotating object like a planet in the vicinity of another massive object like the Sun: the curvature of spacetime due to the latter causes the orientation of the former s rotation axis to change periodically. However, theories are only viable if they withstand rigorous experimental tests. An accurate understanding of GR and its limitations has both scientific and broader technological impacts. GR is used to predict the motion of bodies within the Solar System and the interactions of massive objects in general, and it provides the scientific foundation for Global Positioning Services (GPS). Thus far, all observational data are consistent with GR. However, GR remains incompatible with Quantum Mechanics, the physics of the very small. Modifications to GR therefore merit exploration. The challenge becomes devising experiments that can discern small deviations from GR. Lunar Laser Ranging is one such test: by measuring gravitational phenomena, it can illuminate the degree to which alternative theories can differ from GR, alternative theories which may be compatible with Quantum Mechanics. Lunar Laser Ranging (LLR) tests Einstein s Theory of General Relativity by measuring the transit time of a laser pulse from an observatory to a corner cube retroreflector on the Moon. This transit time gives the Earth-Moon distance with millimeter precision. Successive measurements can constrain the workings of gravity because GR effects manifest themselves periodically in the evolution of this distance as the Earth and Moon both orbit the Sun. LLR proves to be an excellent test of GR because it involves very massive, interacting bodies, which make GR effects more apparent. However, even with such large objects, the effects of GR remain small, on the order 18

20 of a meter or less, compared to the Earth-Moon distance. The high precision enables LLR to discern the subtle effects, or lack thereof, of slight deviations from the current understanding of gravitation. LLR relies on corner cube reflectors on the surface of the Moon. These reflectors employ perpendicular planes to reflect incoming light in the direction from which it came, in this case back to the Earth. The astronauts of the 1969 Apollo 11 mission installed the first retroreflector array on the surface of the moon. Subsequent Apollo 14 and Apollo 15 missions and two unmanned Soviet Lunokhod rovers brought the total to five reflectors that can be utilized in LLR efforts. The MacDonald Observatory began taking data shortly after the Apollo 11 mission and has achieved a precision of 20 cm. Other observatories have joined in the efforts, including the Observatoire de la Côte d Azur (OCA) in By 2005, measurements had reached a precision of 2 cm. In 2006, the Apache Point Observatory Lunar Laser-ranging Operation s (APOLLO) 3.5 meter telescope joined the coalition and has provided the highest level of precision to date, of a few mm. Figure 1-1 shows the evolution of LLR measurement precision over time. APOLLO and OCA remain the most active in LLR measurements today, with APOLLO contributing the most data. A typical transit time from the observatory to the Moon can be obtained by dividing the distance traversed, twice the Earth-Moon distance, by the speed of light, m/s. The Moon s orbit about the Earth has a semi major axis of 385, 000 km, giving a typical transit time of about two and a half seconds. Each millimeter change adds 6.67 picoseconds to the round-trip travel time. APOLLO sends pulses with a 100 ps temporal width. These pulses of 532 nm (green) light contain about photons [8]. However, processes such as beam divergence limit the number of returning photons to very few. The dominant source of divergence is the Earth s atmosphere. Typical atmospheric seeing is about 1 arcsecond, which extends over 2 km of the Moon s surface [8]. In contrast, each reflector array installed by an Apollo mission has an area on the order of 100 square inches, covering a minuscule amount of the beam s area. The diffractive spread from the 19

21 Figure 1-1: The RMS of the residuals between observations and the range model and binned by year quantify the historical precision of LLR measurements. Apollo, which began taking data in 2006, produces range measurements with millimeter level precision. Figure courtesy of Professor James Battat. corner cubes exacerbates this problem as the light returns to Earth. The corner cube reflector causes the beam to diverge by angle θ. of the corner cube and λ is the wavelength of light. θ = λ, where d is the diameter d For a diameter of one inch, θ = radians. At a distance of m, approximately the Earth- Moon distance, this angle translates to a spread of 16 kilometers. With a diameter of 3.5 meters, the telescope receives about times the number of returning photons. The net effect is a factor of reduction such that on average less than one photon returns for each pulse [8]. The detector must therefore be able to detect single photons with high temporal precision. The returning light corresponds to a 19 magnitude 2 source. The full Moon has a magnitude of 13 and results in a background 10 7 times the signal [8]. The suppression of background light through spatial, temporal, and wavelength filtering should allow measurements during full Moon. However, layers of dust on the surface of the corner cube reflector heat up under the Sun s light. The temperature gradient in 2 Magnitudes represent brightness on a logarithmic scale. Larger, positive numbers correspond to dimmer objects. 20

22 Figure 1-2: The figure on the left shows the temporal spread of measurements in the absence of lunar libration convolved with the trapezoidal signature of lunar libration, which can be modeled. The figure on the right shows the spread of photon returns from a local corner cube reflector, with which the trapezoidal signature is convolved. Figure taken from reference [8]. the corner cube reflector creates a spatial gradient in the index of refraction, which increases the exit beam divergence. The index of refraction gradient attenuates the light a factor of ten, while the dust attenuates the light by an additional factor of ten. The signal at full Moon is therefore about 100 times weaker than expected without dust. Reference [9] discusses this problem at length. Lunar libration further complicates the data. Lunar libration refers to the tipping and tilting of the Moon as seen from the Earth and causes an effective tipping and tilting of the reflector arrays. The photons of the return pulse therefore exhibit greater variation in arrival times, creating a spread of 1 ns or 150 mm in the measurements [8]. Lunar libration produces a trapezoidal signature in the measurements that can be modeled. Figure 1-2 juxtaposes the fiducial return, the spread of photon return times from a local corner cube reflector, and the fiducial return convolved with the trapezoidal signature of lunar libration. The uncertainty due to lunar libration can be reduced by statistical means applied to thousands of successive measurements of returning photons. 21

23 The uncertainty of the mean measured distance equals the uncertainty of a single measurement divided by the square root of the number of measurements. A single photon has an uncertainty of about 30 mm, dominated by the lunar libration. Therefore, 900 measurements are required to reduce the uncertainty to 1 mm [8]. The laser sends pulses of light to the Moon at a frequency of 20 Hz. For a photon return of about 0.1 photon per pulse, the detector receives about 2 photons a second. At this rate, 450 seconds or seven and a half minutes worth of photon measurements yield a millimeter level precision. Several factors contribute to variation in the Earth-Moon distance and must be accounted for when isolating gravitational effects. This most prominent variation, on the order of 10,000 km, occurs because the lunar orbit is elliptical with an eccentricity of [8]. The distance varies with respect to the location of the Sun, which interacts gravitationally with both the Earth and the Moon. Other objects in the Solar System also affect the Earth-Moon distance. Jupiter and Venus in particular have the most notable effect, on the order of a kilometer. The propagation of light through the Earth s atmosphere adds the equivalent of two meters to the measured distance, which can be modeled at the sub-millimeter level using measurements of temperature, pressure, and elevation at the site of observation. The Earth also rotates at a rate of 1 km/s at the equator. The APOLLO system clocks must therefore be accurate to 1 microsecond of absolute time because Earth s rotation produces a 0.4 mm change per microsecond [8] in the measured distance. The transit time also includes a Shapiro delay of 25 ns [8] due to the Earth and Sun s gravitational potentials: the light takes additional time to traverse the curved spacetime caused by the masses of the Earth and Sun. LLR does not directly measure the distance between the centers of mass of the Earth and Moon. The laser traverses the distance between the observatory, in New Mexico, to the retroreflectors, on the surface of the Moon. Additionally, the Earth and Moon are not perfect spheres. This is especially true for the Earth, which has oceans and an atmosphere. Careful calculations obtain the desired distance from the measured distance by accounting for the time-dependent locations of the observatory 22

24 and reflector arrays and the non-uniform, non-spherical shapes of the Earth and Moon. Researchers at the Jet Propulsion Laboratory, Harvard-Smithsonian Center for Astrophysics, Leibniz University in Germany, and l Institut de mécanique céleste et de calcul des Ephémérides (Institute of celestial mechanics and ephemeris calculations) in France develop the extensive computational models to accurately fit the data. The periodic nature of GR effects becomes particularly important when distinguishing them from other effects. The Moon s orbit about the Earth has several associated periods, as listed in reference [8]. Within a synodic period of days, the Moon returns to the same position with respect to the Earth and Sun, in line with the two. Determined in part by the precession of the Moon s orbital plane, the draconic period of days refers to the time by which the Moon returns to the intersection point of its orbit and the ecliptic. The Moon returns to the same position with respect to the stars in a day sidereal period, and the Moon returns to its perigee within a day anomalistic period. The most prominent variations in the Earth-Moon distance relate to the Sun s effect on the system and the lunar orbit s eccentricity. The variation due to the lunar orbit s elliptical shape occurs over the anomalistic period. The Sun has a tidal effect on the Earth-Moon system at periods of 31.8 and days [8]. The first period comes from a combination of the effect of the eccentricity and the Sun s perturbation of the Earth-Moon system [11]. The second is equivalent to half the synodic period [11]: the tidal effect is maximized when the Earth and Moon align in the direction of the Sun. Some gravitational effects manifest themselves in the Earth-Moon distance according to the synodic period. Gravitomagnetism, for example, produces signals at both the synodic period and half the synodic period. LLR probes several aspects of gravitation. It can constrain the degree to which the gravitational constant G may be changing, which has implications for the presence of dark energy and the accelerating expansion of the universe. LLR currently limits Ġ to a part in G 1012 per year [5], or about 1% over the age of the Universe. It also 23

25 tests the inverse square law of gravitation. An alternative, the Yukawa potential, has ( both a Newtonian term and a perturbation: U(r) = ) GMm r 1 + αe r/λ. α gives the strength of the perturbation and λ, the scale length. For r much greater than λ, the exponential term approaches zero and the potential behaves like the Newtonian case. For r much smaller than λ, the exponential term approaches one and the potential becomes U(r) = GMm (1 + α). This result is equivalent to scaling the gravitational r constant to G = G(1 + α), which again looks Newtonian. Tests of the inverse square law are most sensitive to the Yukawa potential on scales the size of the experiment with λ r. For LLR, this scale is the Earth-Moon distance. LLR limits a divergence from the inverse square law to about times the strength of gravity at distances of 10 8 meters [1], the most stringent constraint on α at any length scale. LLR also constrains alternate theories of gravitation using the Parameterized Post- Newtonian Framework. The PPN framework consists of a general metric describing spacetime of which GR is a special case. An expansion of this metric in orders of ( v ) 2 c has coefficients β and γ. β quantifies the nonlinearity of gravity, or, that gravity acts on gravitational potential energy, such as the binding energy of an object. The parameter γ quantifies the curvature of spacetime caused by a unit of mass. In GR, β = 1 and γ = 1. Reference [16] determines that β differs from 1 by less than using the most stringent constraint on γ from the Cassini mission: γ differs from 1 by less than [2]. LLR derives upper limits for PPN parameters by constraining gravitational effects. These effects include geodetic precession and gravitomagnetism. The former refers to the precession of an object s rotation axis due to the spacetime curvature engendered by the more massive object that it is orbiting. LLR measures geodetic precession to fall within 0.3% of the GR prediction [7]. Gravitomagnetism, the gravitational equivalent of the magnetic force in electromagnetism, produces two oscillations in the Earth-Moon distance, one with an amplitude of 6.1 meters at the synodic period and another with an amplitude of 6.5 meters at half the synodic period [10]. LLR confirms gravitomagnetism to about 0.15% of the GR prediction [10]. 24

26 LLR also probes the Strong Equivalence Principle (SEP). The SEP states that a gravitational force acts on the gravitational binding energy of an object in addition to its mass. The SEP also states that the gravitational binding energy contributes equally to an object s inertial and gravitational mass, where the first determines an object s resistance to a change in motion and the second determines the strength of the gravitational interaction. The gravitational binding energy or self-energy can be thought of as the amount of energy needed to assemble an object and can be found by summing the potential energy between all mass elements in the object. The Sun, Earth, and Moon have gravitational binding energies, expressed as a dimensionless fraction respectively. U where c is the speed of light, on the order of 10 6, 10 10, and 10 11, Mc 2 SEP. U GM 2 R, so U Mc 2 These values underscore the necessity of massive objects to test the more significant with increasing mass. is proportional to M. The fractional binding energy becomes The Earth s binding energy is approximately twenty-four times that of the Moon. This differential in gravitational self-energy produces measurable differences in the rate at which Moon and Earth accelerate towards the Sun, allowing LLR to constrain violations of the SEP. The dimensionless parameter η quantifies the degree of violation. η equals a linear combination of β and γ, 4β γ 3. In GR, η = 0, while η = 1 maximizes a violation of the SEP. In reference [13], Nordtvedt presents a quasi-newtonian acceleration equation that consists of a Newtonian term and a non-newtonian term, accounting for an SEP violation. In the case of an SEP violation, he predicts a measurable perturbation of the lunar orbit. An SEP violation displaces the Moon s orbit about the Earth towards the Sun if the binding energy contributes more to the inertial mass than the gravitational mass. A violation results in an additional δr meters between the Earth and Moon. This offset varies with the synodic period, the day lunar phase period from new Moon to the next new Moon, corresponding to angular frequency Ω: δr = 13η cos(ωt) meters (1.1) 25

27 If the gravitational binding energy couples more strongly to the gravitational mass, the lunar orbit shifts away from the Sun and δr is maximized at full Moon. Generally, δr reaches its maximum when the three bodies are aligned. A total violation (η = 1) of the SEP would produce maximum offset of thirteen meters. LLR has constrained η to be less than (4.4 ± 4.5) 10 4 [16]. LLR cannot distinguish between the SEP and the Weak Equivalence Principle (WEP), which relates to an object s composition. Rather, LLR relies on laboratory tests of WEP using objects with compositions similar to that of the Earth and the Moon to discriminate between the two effects. A model is fit to the data to obtain values for the GR parameters in addition to other impactful, but uninteresting parameters like the masses of the objects. The work presented here parallels this process, but takes a far simpler approach. Instead of working with actual data, simulations of the Earth-Moon-Sun system generate a data set to which a model can then be fitted. These simulations treat each of the three objects as point masses and ignore the effects of other objects in the Solar System. Neither the simulations nor the fit involve non-gravitational effects like the Earth s atmosphere. Rather, these simulations can be used as a testing ground focused solely on the physics of gravitation. In particular, they can clarify the sensitivity of the Earth-Moon distance to the different parameters of interest, like η, and parameters such as the masses of the Earth, Moon, and Sun. Additionally, since the input values for the simulation are known, the fit can clarify the degree to which one parameter can absorb the effect of another, i.e. the correlations between parameters. The Schwarzschild radius indicates the distance from an object at which the escape velocity from the gravitational potential well equals the speed of light. The curvature of spacetime is most extreme at distances on the order of a Schwarzschild radius from an object, which equals 2GM c 2. The Sun has a Schwarzschild radius of about 3 km. If the entire mass of the Sun were concentrated at its center, an object within 3 km of the Sun s center would be unable to escape the Sun s gravitational pull. The Earth-Sun distance far exceeds the Sun s Schwarzschild radius. Because spacetime is relatively flat at one astronomical unit from the Sun, this test of gravitation falls 26

28 within the weak-field regime. The weak-field regime permits the use of the quasi-newtonian acceleration equation and a first-order multipole expansion of the PPN general relativistic acceleration, described in Chapter 2. These simulations all make use of some form of a position-update method to calculate the objects trajectories. The code calculates the acceleration of each body at their starting positions and updates their velocities and positions over some time step dt: v f = v i + a dt r f = r i + v f dt To increase accuracy, the actual simulations employ a more complicated numerical method, the Fourth Order Runge-Kutta Method, detailed in Chapter 3, which also includes a description of the initial conditions of the simulations. Chapter 4 presents results of the simulations. Chapter 5 describes the method used to fit a model to the data. Chapter 6 summarizes tests of the fit, and Chapter 7 presents initial results of fitting an SEP violation with a Newtonian model. 27

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30 Chapter 2 Underlying Physics 2.1 The Equivalence Principle and Quasi-Newtonian Acceleration A fundamental assumption of GR, the Equivalence Principle (EP) states that objects interact with a gravitational field irrespective of their masses, or that the gravitational mass m g and inertial mass m i of an object are equivalent. The inertial mass determines an object s resistance to a change in motion, while the gravitational mass can be thought of as mass charge, determining the strength of a gravitational interaction. In Newtonian gravity, this assumption allows the cancellation of the masses on both sides of the equation: F g = Mm gg r 2 (2.1) m i a = Mm gg r 2 (2.2) a = MG r 2 (2.3) However, the gravitational self-energy of an object may contribute to its gravitational and inertial mass. The Strong Equivalence Principle (SEP) states that while a gravitational force acts on the gravitational binding energy of an object in addition to the rest of its mass, the gravitational binding energy contributes equally to the gravita- 29

31 tional and inertial masses. In the case of an SEP violation, the gravitational binding energy contributes disproportionately to the gravitational or inertial mass of an object. The mass ratio can then be written as the following, where the dimensionless η parameter indicates the degree of the SEP violation. ( ) m g U = 1 + η m i mc 2 (2.4) Dividing the gravitational self-energy by mc 2 ensures that the right-hand term has no units. The gravitational self-energy can be found by summing the potential energy between all pairs of mass elements throughout the body. dm 1 = ρ( r 1 )d 3 r 1, where ρ is the density as a function of the mass element s position r 1 and d 3 r 1 is a volume element. Similarly, dm 2 = ρ( r 2 )d 3 r 2. The potential energy between the pair of mass elements is G (dm 1)(dm 2 ) r 1 r 2. The total gravitational binding energy becomes the following, while a factor of 1 2 ensures that the pairs are not double-counted. U = G 2 V olume ρ( r 1 )ρ( r 2 ) r 1 r 2 d3 r 1 d 3 r 2 (2.5) In reference [16], Williams includes U mc 2 were obtained by numerical integration of complex models. values for the Sun, Earth, and Moon that ( ) U mc ( 2 ) U mc ( 2 ) U mc 2 s e m = = = The Earth s fractional binding energy is approximately twenty-four times that of the Moon. The relative sizes of their fractional binding energies produce measurable differences in the rate at which they accelerate towards the Sun, allowing LLR to constrain violations of the SEP. 30

32 The gravitational self-energy can also be found analytically for a uniform density sphere, for which U = 3Gm2, derived in Appendix A.1. For the Earth and Moon, 5r this equation gives: ( ) U mc ( 2 ) U mc 2 e m = = The above values have a 10 and 1.2 percent difference, respectively, from those obtained by numerical integration. Incorporating Equation 2.4 into Equation 2.3 results in a quasi-newtonian representation of the acceleration that accounts for an SEP violation. a = m g MG (2.6) m i r [ 2 ( )] U MG a = 1 + η (2.7) mc 2 r The Nordtvedt Effect In reference [13], Nordtvedt predicts a polarization of the lunar orbit if the SEP is violated. Using the gravitational binding energy of a uniform sphere, the ratio of the gravitational and inertial masses can be written as: m g m i = 1 + η ( ) 3Gm 5Rc 2 (2.8) The gravitational binding energy contributes disproportionally to the gravitational mass, which becomes greater than the inertial mass. For the Earth, the small, additional acceleration, the right-hand term from equation 2.7, becomes the following, where r es is the Earth-Sun distance: ( ) 3Gme GMs δa = η 5R e c 2 r 2 es (2.9) 31

33 A term with the above form can be added to the Moon s equations of motion, describing a circular orbit about the Earth, to reflect the additional acceleration of the Earth towards the Sun. φ represents the angle between the Moon s position and the line between the Sun and Earth and relates to the phase of the Moon. φ = Ωt, where Ω is the synodic frequency. Below, µ = Gm e, h is the specific angular momentum of the Moon, and r denotes the Earth-Moon distance. d 2 r = h2 dt 2 r µ + δa cos(φ) (2.10) 3 r2 dh = δa r sin(φ) (2.11) dt Appendix A.2 reproduces the full derivation, which follows from the equations above by writing r as r 0 + δr and h as h 0 + δh. Substitutions using µ = ω r 0, 2 where ω is the orbital frequency of the Moon about the Earth and corresponds to the anomalistic period; the specific angular momentum of the Moon h 0 = r 2 0ω 0 ; and ω 2 0 Ω 2 = 2 13 ω2 0 lead to the following result. δr = η r3 GM s cos(ωt) (2.12) c 2 R e res 2 An SEP violation results in a polarization in the lunar orbit, with an additional δr meters between the Earth and Moon. Using the accepted values of r es = meters, R e = meters, M s = kg, and r = meters, the equation simplifies to 1 : δr = 6.9η cos(ωt) meters (2.13) δr is maximized when the three bodies are aligned. δa and therefore δr are in fact proportional to the difference in the gravitational binding energies. Nordtvedt neglects the binding energy of the Moon, an order of magnitude smaller than that of the Earth. He also sets the synodic period equal to 1 In reference [13], Nordtvedt mistakenly uses a coefficient of 6 5 energy of a sphere. His final result is a factor of two off. for the gravitationally binding 32

34 the anomalistic period. In the absence of Nordtvedt s approximations, per reference [15], the equation is: δr = δr = ( ω 0 ) Ω ηδa cos(ωt) (2.14) ω0 2 Ω ( ω 0 ) ( Ω GMs Ue ω0 2 Ω 2 m e c U ) m η cos(ωt) (2.15) 2 m m c 2 r 2 es Reference [15] also states that ω 0 = 13.4ω s = s 1, where ω s is the angular velocity of the Earth about the Sun, and that Ω = ω 0 ω s. Using these values, the equation becomes: ( δr = Ue m e c U ) m η cos(ωt) meters (2.16) 2 m m c 2 For the numerically obtained values of the gravitational binding energies, this equation yields an amplitude of 8.0 meters. For uniform spheres, the amplitude is 7.1 meters. References [12] and [3] find that the Sun s perturbation of the Earth-Moon system amplifies the Nordtvedt Effect such that the coefficient of meters increases to about meters. This amplification increases the polarization to about 11.5 meters for uniform spheres and 12.9 meters for numerically obtained gravitational binding energies. 2.2 General Relativistic Acceleration The Parameterized Post-Newtonian (PPN) framework encompasses a broad range of gravitational theories, including GR, by positing a general metric and expanding it in powers of ( v c ) 2. Per references [4] and [14], the first-order expansion of the 33

35 acceleration, valid in the weak-field regime, is: a A = ( GM B ( r B r A ) 1 r 3 B A AB ( va ) 2 ( vb γ + (1 + γ) c c + 1 c 2 B A 2(β + γ) c 2 C A,B ) 2 2(1 + γ) c 2 v A v B 3 2c 2 GM C 2β 1 r AC c 2 C A,B ( ( ra r B ) v B r AB GM C r BC + GM B (( r rab 3 A r B ) ((2 + 2γ) v A (1 + 2γ) v B ))( v A v B ) γ 2c 2 ) c 2 ( r B r A ) a B The Newtonian acceleration can be used for that of the second body, a B, on the right-hand side of the equation above [4]. For β = γ = 1, the equation reduces to the GR acceleration: ( va c ) ( vb c a A = B A + 1 c 2 B A GM B ( r B r A ) r 3 AB ) 2 4 c 2 v A v B 3 2c 2 ( 1 4 c 2 C A,B ( ( ra r B ) v B r AB GM C 1 r AC c 2 C A,B B A GM C r BC + ) c 2 ( r B r A ) a B GM B (( r rab 3 A r B ) (4 v A 3 v B ))( v A v B ) + 7 GM B a B 2c 2 r AB B A The first term GM B(r B r A ) is the Newtonian component in the acceleration, while rab 3 the remaining terms comprise the relativistic contribution to the acceleration. The general relativistic acceleration accounts for all GR phenomena such as gravitomagnetism, or in the case of β 1, γ 1 case, non-gr effects. ) GM B a B r AB ) 34

36 Chapter 3 Simulation Characteristics 3.1 Numerical Method Mentioned in the Introduction, the Euler method gives a straightforward prescription to update the velocity and position of an object, where dt is a small, finite time step. v f = v i + a dt r f = r i + v f dt First, the acceleration a of the object in question must be found by considering its gravitational interactions with the other two objects. For example, in the Newtonian and quasi-newtonian case, the total acceleration of the Earth equals the sum of the accelerations due to the Moon and Sun. The acceleration is a function of the initial positions since the strength of a gravitational interaction depends on the distance between objects. The direction of the acceleration also depends on the position vector from one object to the other. Updating Equation 2.3 to account for the position vectors and the multiple objects, the acceleration of the first object becomes the following, where r AB denotes the vector from A to B. a 1 = M 2G r 2 12 ˆr 12 + M 3G r 2 13 ˆr 13 = M 2G r r 12 + M 3G r r

37 The calculated acceleration is used to update the velocity, which is then used to update the position. This method must be applied to each of the three objects. The more involved Fourth Order Runge-Kutta method applies to a function r whose derivative with respect to t depends on itself and t: dr dt = f(t, r). The rate of change or slope of r is recalculated four times using f(t, r) over the interval dt. This method takes a weighted average of the slopes m to update r. 1 m 1 = f(t, r) ( m 2 = f t dt, r + 1 ) 2 m 1 dt ( m 3 = f t dt, r + 1 ) 2 m 2 dt m 4 = f(t + dt, r + m 3 dt) r final = r initial (m 1 + 2m 2 + 2m 3 + m 4 )dt Updating the position of an object involves both a first and second order derivative: acceleration equals the rate of change of velocity and velocity equals the rate of change of the position. The steps outlined above must be repeated for both the velocity and position. Acceleration, the second derivative of position, depends only on position. r 1 and v 1 denote the initial position and velocity, respectively. a 1 = f( r 1 ), where f( r) calculates the gravitational acceleration at some position r and has no timedependence. 1 Professor Erik Cheever in the Swarthmore College Department of Engineering provides educational materials on the Fourth Order Runge-Kutta Method at 36

38 The calculations proceed as follows: ( ) 1 v 2 = v 1 + a 1 2 dt ( ) 1 r 2 = r 1 + v 2 2 dt a 2 = f( r 2 ) ( ) 1 v 3 = v 1 + a 2 2 dt ( ) 1 r 3 = r 1 + v 3 2 dt a 3 = f( r 3 ) v 4 = v 1 + a 3 (dt) r 4 = r 1 + v 4 (dt) a 4 = f( r 4 ) Finally, v f = v i + 1( a a a 3 + a 4 )dt, r f = r i + 1( v v v 3 + v 4 )dt, and t f = t i + dt. The process repeats until t f equals the desired simulation duration. A plot of the system s energy over time gauges the accuracy of these numerical methods. The system of the Earth, Moon, and Sun, treated as point particles, lacks dissipative forces. Therefore, the total energy is conserved. After each iteration of the velocity and position update method, the total energy of the system at time t can be found by summing the potential and kinetic energies of the bodies. For the first object, the total energy is: U = Gm 1m 2 r 21 Gm 1m 3 r 31 K = m 1v E = K + U The system s total energy equals the sum of the kinetic and potential energies for all 37

39 Figure 3-1: The system s energy appears to be conserved for four-year long simulations using the Euler method (left) and Fourth Order Runge-Kutta method (right). Both exhibit fluctuations in the energy, but neither has a net gain or loss of energy over the course of the simulation. The fluctuations are small for both cases. For the Euler method, the energy fluctuations are a part in 10 5 change in the total energy. The Fourth Order Runge-Kutta method is more optimal: the energy fluctuations are a part in 10 9 change in the total energy. the objects. The energy should exhibit very little fluctuation, remaining relatively constant with time. Figure 3-1 presents a comparison of the energy versus time for the Euler method (left) and the Fourth Order Runge-Kutta method (right). The energy of the system fluctuates for both cases. For the Euler method, the energy fluctuations have an order of magnitude of for a total energy on the order of Joules, a part in 10 5 change. The Fourth Order Runge-Kutta reduces these periodic fluctuations by another four orders of magnitude, confirming that it is the superior method. In both cases, the energy of the system remains relatively constant over the four years. That is, despite fluctuations, the system does not exhibit a general trend over time of a net gain or loss in energy. 3.2 The Time Step The time step can have a significant impact on the accuracy of the simulation because the acceleration is treated as constant over the time step when it actually changes instantaneously with position. A small enough time step must be chosen such that 38

40 the acceleration remains nearly constant over the small change in time and distance, from r to r + r. A plot of the system s energy versus time clarifies the impact of the time step on a Fourth Order Runge-Kutta four-year long simulations of both the Newtonian and SEP violation cases. Figure 3-2 juxtaposes four such plots for time steps of 2000, 1000, 500, and 100 seconds. Unsurprisingly, the fluctuations decrease in magnitude with smaller time step. The energy fluctuation decreases by an order of magnitude from 2000 seconds, with E Joules, to 1000 seconds, with E Joules, and again from 1000 seconds to 500 seconds, with E Joules. Ten orders of magnitude below the system s energy for time steps on the order of 100 seconds, the fluctuations are negligible. The 2000 second time step simulation exhibits a net loss in energy with time. The maxima of the fluctuations fall by Joules over the course of the simulation. Most simulations employ either a 1000 or 500 second time step since the fluctuations are at least nine orders of magnitude smaller than the total energy. The dt = 100 seconds simulation is avoided for practical reasons: the code takes six times as long to run. The full SEP violation model appears to be less sensitive to reductions in the time step. Its fluctuation E remains about Joules greater than the E of the Newtonian simulation regardless of the time step. Figure 3-3 highlights this phenomenon by plotting the system s energy for both models at time steps of 1000, 500, and 100 seconds on the same axes. This phenomenon may occur because the acceleration in an SEP violation is larger than the Newtonian acceleration by η U mc 2. Used to calculate the energy of the system, the velocities and positions evolve slightly differently in the SEP violation case than in the Newtonian case. The SEP violation total energy may be susceptible to different numerical errors than the Newtonian case at all time step sizes. 3.3 Initial Conditions and Assumptions The simulation makes several simplifying assumptions. It treats each object as a point mass so that the simulated distance is the distance between the centers of mass 39

41 Figure 3-2: The figures above plot the total energy of the system versus time for time steps of 2000 seconds (upper left), 1000 seconds (upper right), 500 seconds (lower left), and 100 seconds (lower right). The total energy corresponding to the 2000 second time step exhibits a noticeable net loss in energy over the four-year long simulation. The energy fluctuates slightly in each case. In the order of decreasing time step, the fluctuations E are 10 25, 10 24, 10 23, and Joules. of the Earth and Moon; the code does not need to account for the location of the observatory. This simplification also eliminates the need to consider the complex and time-variable shapes of the Earth and Moon. The results in Chapter 4 use the numerically obtained values for the gravitational binding energies of the Earth, Moon, and Sun to compare with the 13 meter theoretical prediction. Chapter 4 also summarizes results using the gravitational binding energy of uniform density spheres for the Earth, Moon, and Sun. While unrealistic, this assumption becomes necessary for the data fitting portion of this project, where changes in the mass of an object must be reflected in its gravitational binding energy. Simply inputting the values obtained by complex models from the literature would be effectively treating the 40

42 Figure 3-3: The system s energy versus time is plotted for four-year Newtonian and quasi-newtonian (η = 1) simulations using time steps of 1000, 500, and 100 seconds. The energy fluctuations for the Newtonian case are always Joules less than the energy fluctuations of the full SEP violation case even as the size of energy fluctuations decreases for both models. binding energies as constants. The initial conditions place the Sun, Earth, and Moon in a line along the x-axis in that order. At first, the Sun is placed at the origin. Since the Sun-Earth separation is about meters, the Earth s position r e equals ( , 0) meters. The approximate Earth-Moon distance is taken to be meters. The Moon s initial position becomes the sum of the vectors from the origin to the Earth and from the Earth to the Moon: r m = ( , 0) meters. The code then locates the system s center of mass given these conditions and adjusts the initial position vectors of each body to place the center of mass at the origin. Each object has only tangential velocity, in the ŷ direction, at the start of the simulation. The initial velocity of the Earth is taken to be v e = (0, ) m/s since the approximate speed of the Earth about the Sun is m/s. Since the Moon s speed about the Earth is approximately m/s, its initial velocity is taken to be the v m = (0, ) m/s. The code calculates the Sun s velocity to ensure that the center of mass remains stationary and finds that it must be v s = (0, ) 41

43 Figure 3-4: Not to scale, the cartoon depicts the initial conditions of the simulation, which places the Sun, Earth, and Moon along the x-axis in the center of mass frame. Arrows denote initial velocities. m/s in the center of mass frame. Figure 3-4 provides a cartoon representation of the simulation s initial conditions. Table 5.1 gives a complete list of the initial conditions after the center of mass has been fixed at the origin. To check that the center of mass remains fixed at the origin, its position and velocity can be calculated during each time step of the simulation. The following calculations can be inserted into each iteration of the update method. p~s + p~e + p~m ms + me + mm ms~rs + me~re + mm~rm = ms + me + mm ~vcm = ~rcm Using the Fourth Order Runge-Kutta method, Figure 3-5 illustrates the center of mass s behavior during the four year duration of a simulation with a time step of 1000 seconds. The center of mass remains within meters of the origin for the entire simulation, and its speed never exceeds m/s. The center of mass drifts slightly due to small numerical errors that accumulate over the simulation. Since LLR measures the Earth-Moon distance to a few millimeters precision, and since the work presented here focuses on effects on the order of meters, the center of mass is regarded as stationary at the origin for the purposes of these simulations. 42

44 Figure 3-5: The velocities and positions of the Earth, Moon, and Sun are adjusted so that the center of mass remains stationary at the origin. Small numerical errors give the center of mass a non-zero position and velocity, which then changes slightly during the simulation. The coordinates for the center of mass position and velocity versus time are plotted above. While non-zero, both the velocity and position coordinates remain negligible, on the order of m/s and 10 8 meters, respectively. 3.4 Precision To clarify the effects of precision on the simulation results, two simulations with different levels of precision were compared. All simulations presented thus far have utilized Numerical Python s np.float64 option, which represents numbers using 8 bytes, or 64 bits. A three-year long simulation with a time step of 500 seconds was compared with an identical simulation with increased precision using the np.float128 option. Figure 3-6 presents the differences in the generated Earth-Moon distances for these simulations. The differences are minimal: they never exceed four centimeters. 43

45 Figure 3-6: Two three-year long simulations, one using Python s np.float64 option and the other, np.float128, were differenced to determine the effect of precision on the computed Earth-Moon distances. The differences between the two simulations were minimal. The distances never differed by more than four centimeters. 44

46 Chapter 4 Simulation Results Figure 4-1 presents the Earth-Moon distance over the course of two years using the Fourth Order Runge Kutta method and a time step of 500 seconds. It includes distances computed using a Newtonian model and distances computed using a quasi- Newtonian model with η = 1, maximizing the SEP violation. The average Earth- Moon distance over a two year simulation equals kilometers, which differs from the value of kilometers in reference [10] by about 4000 kilometers, or 1%. The simplified initial conditions of these simulations result in a slight divergence of the average Earth-Moon distance from the accepted value. The distance ranges from approximately to kilometers. Reference [10] states that the Earth-Moon distance changes by about kilometers, primarily due to the elliptical shape of the Moon s orbit about the Earth. The Earth-Moon distance varies by about kilometers in these simulations, about a factor of three smaller than the accepted range. This difference implies that the Moon s orbit in these simulations has a lower eccentricity than the actual orbit. 45

47 4.1 Comparing Newtonian and quasi-newtonian Simulations The differences between the Newtonian and quasi-newtonian simulations are not discernible in a simple plot of the distances. However, they become more apparent if the former data set is subtracted from the latter. Figure 4-2 shows both a plot of the distances and a plot of the difference between the Newtonian and full SEP violation distances for a duration of four years. The result has a packet form, characteristic of a superposition of periodic changes in the distance. The shape of the curve therefore indicates that there are multiple periodic signals in the distance differences of the simulations. The envelope dissociates slightly with time, while it also increases in amplitude. The disintegration of the envelope structure becomes more apparent in the differences of the Newtonian and SEP violation seven-year long simulations in Figure 4-3. The dissociation of the envelope and growth in amplitude may be due in part to error accumulated over the duration of the simulation. For example, it may be a result of the assumption the acceleration remains constant over the time step. However, seven year simulations using a time step of 1000 and 200 seconds yielded the same results, implying that the shape of the curve reflects the physics of the system rather than numerical error. The increasing amplitude and disintegration of the envelope structure suggest that the simulations diverge from one another with time, an expected result since the acceleration equations differ very slightly according to the value of η, which will cause the velocities and positions to evolve differently. For example, the Moon s orbit precesses at a rate that depends on the degree of SEP violation, or lack thereof. Since the difference between the Newtonian and quasi-newtonian accelerations is very small, a consequence of the ratio of gravitational binding energy to Mc 2, the growth of the envelope s amplitude is very gradual: it increases by about five meters over seven years. The divergence of the simulations casts doubts on results obtained by directly subtracting one simulation from another. 46

48 Figure 4-1: The Earth-Moon distance for a two-year long simulation with a time step of 500 seconds exhibits a variety of periodic behavior that depends on the the positions of all three bodies with respect to each other. The average Earth-Moon distance is about kilometers, and the distance varies from about to kilometers. 4.2 The SEP Violation Signal A discrete Fourier transform of the differences between the simulations can be used to isolate the different periodic signals. The difference between the simulations can be represented as a sum of sinusoids, each with their own period and amplitude. The Fourier transform determines the frequencies of the signals and the sizes of their contributions to the differences in the form of an amplitude. These signals appear as peaks in the Fourier spectrum at the contributing frequencies. The height of the peak gives the amplitude of the signal. The frequencies, in units of s 1, can be converted to periods in units of days to understand the origins of each signal. As is characteristic of a Fourier transform, the width of the signals in frequency space is inversely related to the duration of the simulation. That is, the simulation must have a long enough duration to resolve the signals in frequency space. Figure 4-4 corresponds to a four-year long simulation, while Figure 4-5 corresponds to a seven- 47

49 Figure 4-2: The first plot shows the Earth-Moon distance over the four-year long simulations with η = 0 and η = 1. The lower plot consists of the differences between those two simulations, r(η = 1) r(η = 0). The differences have a wave-packet form that begins to disintegrate towards the third year. year long simulation. These simulations continue to use the numerically obtained values for the gravitational binding energies of the objects. The seven-year long simulation peaks have about half of the width of those in the four-year long simulation spectrum. The seven year spectrum also resolves the right-most peak, and the middle two peaks between periods of 25 and 30 days are distinct. In anticipation of fitting the Earth-Moon distances, Figure 4-6 presents the results for a two-year long simulation. Since fitting the data is a much more involved and time-consuming process, reducing the duration of the simulations to two years will allow the fit code to finish running in a timely manner. However, a shortened duration will also mean reduced resolution 48

50 Figure 4-3: r(η = 1) r(η = 0) for seven-year long simulations exhibits an almost identical form to that in Figure 4-2, for four-year long simulations. However, as the simulations progress, the wave packet continues to dissociate due to the slight differences in the accelerations, which, for example, impact the precession of the lunar orbit. in the Fourier spectrum. Figure 4-6 shows that while the widths of the peaks has increased such that they overlap, the adjacent peaks around 28 days remain distinct. Nordtvedt predicts a signal of amplitude about 13 meters at the synodic frequency for an SEP violation with η = 1. The signals in Figure 4-4 and Figure 4-5 fall at a slightly shorter periods of about and days, compared to the actual synodic period of days. The peaks in the four and seven year long simulations have amplitudes of 11 and 12.5 meters, respectively. The longer simulation gives an amplitude of the signal closer to the expectation of about 13 meters. The spectrum also features three other prominent signals. Reference [10] lists several periodic effects the Earth-Moon distance. The elliptical orbit of the Moon results in a signal at days, the anomalistic period. The Sun s 49

51 Figure 4-4: The Fourier spectrum of r(η = 1) r(η = 0) for four-year long simulations exhibits four prominent signals with periods corresponding to the the Nordtvedt effect (29.53 days), the Sun s perturbation of the Moon s orbit (31.8 days and days), and the elliptical shape of the lunar orbit (27.55 days). The periods of the signals in the spectrum are slightly less than the accepted values. The SEP violation signal has an amplitude of approximately 11 meters, slightly less than the 13 meter prediction. interaction with the Earth-Moon system causes oscillations in the distance at periods of 31.8 days and days. The other three peaks in the Fourier spectrum relate to these effects. In Figure 4-5, the signals occur at days with an amplitude of 4.46 meters, days with an amplitude of meters, and days with an amplitude of 1.68 meters. Because the quasi-newtonian acceleration is larger than the Newtonian acceleration, the velocities and positions of the objects and therefore the lunar orbit evolve differently. As a result, η impacts the Sun s interaction with the Earth and Moon system. The amplitudes associated with the Sun s perturbation of the Earth-Moon system differ very slightly by 4.46 and 1.68 meters for η = 1 and η = 0. Similarly, for even slight differences in the accelerations of the Earth and Moon, the eccentricity of the Moon s orbit about the Earth changes. The oscillation in the Earth-Moon distance associated with the elliptical shape of the lunar orbit differs by meters for η = 1 and η = 0. As was the case for the SEP violation 50

52 Figure 4-5: Increasing the duration of the simulation from four to seven years increases the resolution of the Fourier Spectrum. The signals also increased in amplitude, with the SEP violation signal having an amplitude of 12.5 meters, closer to the predicted 13 meters. signal, these signals in Figure 4-5 occur at periods that are slightly smaller than the accepted values given in reference [10], perhaps a result of the simulations simplified initial conditions. The SEP violation signal in the two-year spectrum has an amplitude of about 12.9 meters and a period of 29.2 days. While these values are the closest to the expectations of 12.9 meters and days, the resolution in frequency space leaves these results open to question. The signal appears to decrease and then increase in amplitude with increasing duration. Similarly, the period shifts towards 28.8 days, the period of the signal in the three year spectrum, and then back towards 29 days. These fluctuations appear to persist for durations up to at least 15 years documented in Appendix C. The fluctuations in the measured period and amplitude result from the sampling in frequency space and the method by which the values were obtained. The measured amplitudes and periods as presented here come from that of the highest point in the 51

53 Figure 4-6: This figure presents the results from differencing two-year long SEP violation and Newtonian simulations. The resolution of the Fourier spectrum suffices to distinguish the SEP violation signal and the signal associated with the eccentricity of the lunar orbit. The amplitude (12.9 meters) and period (29.2 days) of the Nordtvedt Effect signal best match the expectations of days and 12.9 meters. peak. However, the spectrum consists of discrete points, and most likely the highest point does not fall at the maximum of the signal s profile. Figure 4-5 gives a more accurate amplitude by this method because the symmetric distribution of points in the SEP violation signal results in one of them falling very near the center of the peak, where the maximum would also fall. In contrast, no point in the signal of Figure 4-4 falls at the center of the peak. Properly deriving values for the period and amplitude entails fitting a Gaussian to the signal, which would also clarify the cause of the variation. If the inconsistencies were to persist even with the more optimal method, they would indicate a subtler flaw in directly subtracting simulations. The SEP violation and Newtonian simulations diverge with time, rendering results obtained by direct subtraction subject to question. As mentioned earlier in this chapter, the rate at which the Moon s orbit precesses depends on η and may effect the results when directly subtracting simulations. The Nordtvedt Effect is minimized and maximized when the Earth, Moon, and Sun are aligned, or when the Moon is at new or full Moon. δr is therefore proportional to the cosine of the lunar phase angle. Setting the phase equal to the synodic frequency multiplied by time assumes that the synodic period is unaffected by an SEP violation. 52

54 However, a slight change in the accelerations of the objects will change the synodic period very slightly, which will in turn affect results more apparently with increasing duration. Fitting the simulated data should alleviate this problem. An alternative method of isolating the SEP violation signal involves fitting η = 1 data with a Newtonian model, with η frozen at zero. The fit will adjust parameters to absorb the periodic signals, but it should be unable to absorb the full 13 meters of the SEP violation signal, which should then dominate the residuals between the final model and the original data. Chapters 5 through 7 explore this possibility. The fit will treat the Earth, Moon, and Sun as uniform density spheres. A two-year long simulation using the analytic gravitational binding energies gives an amplitude of about 12 meters, slightly higher than the 11.5 meters calculated in Chapter 2. The associated period was about 29.2 days. This amplitude corresponds to about a four percent increase in the coefficient in Equation Reference [3] finds the coefficient in Equation 2.16 to be meters, which yields an amplitude of about 11.7 meters. Following Nordtvedt s example of dropping the Moon s gravitational binding energy, a twentieth of that of the Earth, predicts an amplitude of 12 meters. Assuming the lessened resolution has no significant effect on the amplitudes of the peaks, the result remains within an acceptable range of the expectation. The simplified initial conditions may affect the amplitude and period of the SEP violation signal. When the initial Earth-Moon distance was reduced from meters to meters to match the initial conditions used in the fit, the amplitude of the signal decreased by 0.2 meters and the period shifted to days. The specific impact of each of the simplifying assumptions of these simulations is reserved for future work. 53

55 Figure 4-7: The amplitude of the envelope appears to be proportional to η. The figure above shows both r(η = 1) r(η = 0) and r(η = 0.5) r(η = 0) versus time. The latter has half the amplitude of the former, which the Fourier spectrum further emphasizes. The signals in the Fourier spectrum for the η = 0.5 case have half the amplitude of those for the η = 1 case. 4.3 The Amplitude of the Envelope and its Relation to η The amplitude of the envelope appears to be proportional to η. Figure 4-7 plots the differences between a Newtonian and an η = 1 quasi-newtonian simulation in addition to the differences between a Newtonian and an η = 0.5 quasi-newtonian simulation. The amplitude of the latter s envelope is about half the amplitude of the η = 1 envelope. The Fourier spectrum keeps with this finding: the SEP violation signal for η = 1 has double the amplitude of the signal for η = 0.5. This relationship also holds for the other signals in the spectrum. The dependence of the envelope s amplitude on η is consistent with Nordtvedt s assumption that the polarization of the lunar orbit is proportional to η. This discussion has particular relevance to fitting the data, during which the differences between the distances with η = η 0 and η = η 0 + η is proportional to η. 54

56 4.4 General Relativistic Acceleration Another code implements a Fourth Order Runge-Kutta simulation using the PPN acceleration with β and γ described in Chapter 2. It uses a similar method as the SEP violation and Newtonian distances comparison: one simulation is subtracted from the other and a Fourier transform is taken of the difference. Figure 4-8 presents a comparison of a GR simulation with β = 1 and γ = 1 to a Newtonian simulation. While subtracting one simulation from another is not an ideal method of comparison, the Newtonian and quasi-newtonian cases at least differ by only a small perturbation. This method of directly subtracting one simulation from the other will prove more problematic for a GR and Newtonian simulation, which differ by more than a small perturbation to the equations of motion. However, the differences, which exceed meters, seem even more dubious than can be explained by the shortcomings of the method. The implementation of the acceleration equation requires additional vetting before any conclusions can be drawn from these results. The differences exhibit a similar envelope shape to the difference between SEP violation and Newtonian distances. However, the envelope disintegrates more quickly over this six-year long simulation than it did in the SEP violation comparison. Approximately meters, the increase in the envelope s amplitude is both more extreme and more rapid. Future work may investigate whether the fact that the differences between GR and Newtonian simulations are about times the differences between the quasi-newtonian and Newtonian simulations is a coincidence or indicative of a particular error in the implementation. The Fourier spectrum features signals at the same periods as that of the differences between the SEP violation and Newtonian data. These signals should indicate GR effects like gravitomagnetism, but the current amplitudes provide little clarity on which signal corresponds to which effect. After the code has been scrutinized for sources of error, the spectrum may be interpreted for different GR effects. 55

57 Figure 4-8: The difference between six-year long GR and Newtonian simulations has the same envelope form as the other results in this chapter. The amplitude of the envelope and the signals in the Fourier spectrum seem less credible. The amplitudes most likely indicate an error in the implementation of the simulation. 56

58 Chapter 5 Fitting the Simulated Data Fitting the simulated SEP violation data set with a Newtonian model represents a more optimal alternative to subtracting one simulation from the other. This code employs a linear least-squares fit to minimize the residuals between the model and the simulated data. The fitting process begins with a simulated data set, which consists of the Earth- Moon distances and their corresponding times. Called the observed distances, these values represent the measurements of the Earth-Moon distance made by LLR. The parameter values used to simulate these distances are called the observed values. The code also requires a model with which to fit the data. The model assumes values for the initial conditions in addition to a gravitational framework, e.g. a Newtonian model. To properly adjust the model parameters to reflect the observed distances, the code must account for the dependence of the distance on a given parameter. Often, this process would involve finding the partial derivatives of an equation for the Earth-Moon distance with respect to each parameter. However, the method used to calculate the distances lacks a single, comprehensive equation that can be differentiated for every parameter. No analytic solution exists. Instead, the partials of the distance must be found numerically as a finite difference. The code calculates the amount by which a small change in a parameter will change the Earth-Moon distance over the course of a simulation. It generates distances for the initial parameter and for the initial parameter plus the small change. It then subtracts the two data 57

59 sets to obtain the small, finite differences in the r values, the Earth-Moon distances. Similar to partials except finite, the computed differences in r provide information on that parameter s impact on the Earth-Moon distance over that small range of parameter values. The dependence of the distance on that parameter outside of that range of values remains unknown; the partials may or may not be a good predictor of the dependence for other values of that parameter. For parameters on which the distance depends linearly, the partials remain the same for all values of the parameter. Consider a distance x which depends on some variable p such that x = c 1 p + c 2 where c 1 and c 2 are constants. A small change in x, x, is proportional to a small change in the parameter. x p = c 1, which has no dependence on the value of the parameter. x = x p is therefore also independent of p. p However, the Earth-Moon distance does not depend linearly on most of the parameters. So long as the difference between the observed and model value remains on the order of the change used to generate the partials, or finite differences, the non-linearity should not pose a problem. To fit an observed value that differs more substantially from the model parameter s initial value, the fit would need to iterate over many small parameter adjustments from p model to p observed until the fit converges. Since this work focuses on detecting an SEP violation, the observed and initial model values for all parameters except η are equivalent. Chapter 6 explores the effects of fitting an observed η that differs from the model η by a value much greater than the η used to calculate the finite differences. The code proceeds to adjust the model s initial parameters to minimize the differences, or residuals, between the model and the observed distances. The resultant fitted values for the input parameters, called the post-fit computed values, can be used to simulate yet another data set, the post-fit computed distances. The observed distances minus the post-fit computed distances give the post-fit residuals. The differences between gravitational frameworks should manifest themselves in these residuals. In the case of quasi-newtonian data fitted by a Newtonian model with η fixed at zero, even as the fit adjusts parameters to account for the differences between the observations and the Newtonian model, it should fail to absorb the sinusoidal signature of an 58

60 SEP violation. The 12 meter sinusoid should appear as the sole signal in the post-fit residuals. 5.1 Perturbing the Parameters Because the SEP violation oscillation has an amplitude of about 12 meters, changes in parameters are chosen to produce a maximum r of about a meter during the duration of the simulation. Some parameters had r values that increased with time and reached one meter at the end of the simulation, while other p j resulted in periodic behavior in the r values. Table 5.1 presents the fractional and absolute changes in each parameter that change the Earth-Moon distance by one meter during a two-year long simulation with a time step of 1000 seconds. Corresponding parameters for the Earth and Moon require very similar changes to produce a one meter offset in the Earth-Moon distance. For example, both m m and m e are adjusted by an absolute change of approximately kilograms, and both v ey and v my are adjusted by approximately m/s. The reason for these similarities is quite intuitive in some cases. Both r ey and r my are adjusted by 0.75 meters because they have equivalent effects on the initial Earth-Moon distance. Both the Earth and Moon lie on the x-axis. Shifting one of them by 0.75 meters in the y-direction lengthens the initial Earth-Moon distance d 0 to d This idea can be generalized: because both the Earth and Moon s parameters have direct and equivalent relationships to the measurements in question, measurements of the Earth-Moon distance, similar changes in their parameters result in similar offsets. The parameters associated with the Sun, in contrast, require a much greater change to produce a comparable offset because it does not have as immediate an effect on the Earth-Moon distance. 59

61 Parameter Value Fractional Change Change m e m m m s v ex v mx v sx v ey v my v sy r ex r ey r sx r sy r mx r my η Table 5.1: The table presents the nominal values for all the parameters after the center of mass has been fixed to the origin. The table also includes the fractional, when applicable, and absolute change in the parameters to produce a maximum r of one meter over a two-year long simulation. The masses are in units of kilograms, velocities in meters per second, and position coordinates in meters. 60

62 5.2 Fit Method The fitting process first requires a set of simulated observed distance measurements using a particular gravitational framework and initial conditions. The fit code generates a data set using a particular model and initial conditions, which may or may not differ from those used to simulate the observations. In the following discussion, the subscript Ci denotes pre-fit (initial) computed parameter values and distances. The subscript O denotes observed parameter values and distances, that is, the simulated measurements, and the subscript Cf indicates post-fit (final) computed parameter values and distances. The observed distances r O (t) can be written as a sum of the pre-fit computed distances r Ci (t) and small changes. These small additions to the distances correspond to small changes in the parameter values. r O (t) and r Ci (t) are both arrays with m elements, determined by the duration of the simulation and the time step. For n total parameters, n r O = r Ci + b j r j (5.1) j=1 r O = r Ci + b 1 r 1 + b 2 r b n r n (5.2) The r j values are found for each parameter p j by running a simulation with a slight adjustment to the jth parameter and taking the difference with the model distances. For some parameter p j, a change of p j changes the distance measurements by r j, a column vector. The r j form a set of basis functions { r 1, r 2,... r n }. Multiplied by the corresponding r j array, the coefficients b j determine the contribution of each p j to eliminate the differences between the pre-fit computed and observed distances. Another array D can be defined as the difference between the simulated data and the initial model distances, or the pre-fit residuals: D = r O r Ci (5.3) Additionally, the basis functions r j comprise the n columns of a matrix r, a 61

63 rectangular m n matrix. m is the number of observations, equal to the duration of the simulation divided by the time step assuming uniform sampling, and n is the number of model parameters. r me (t 1 ) r mm (t 1 ) r ms (t 1 )... r η (t 1 ) r me (t 2 ) r mm (t 2 ) r ms (t 2 )... r η (t 2 ) r = r me (t 3 ) r mm (t 3 ) r ms (t 3 )... r η (t 3 ) r me (t M ) r mm (t M ) r ms (t M )... r η (t M ) Therefore, D = r B, where B denotes the columnar array of coefficients b i. The fit involves finding B, the coefficients, because they give the weighting each of r j to go from the pre-fit model distances to the observed distances. Solving for B would usually involve finding the inverse of r: B = r 1 D. However, r is a rectangular matrix that needs to be manipulated into a square shape. Multiplying both sides by the transpose of r gives r T D = r T rb. To isolate B, multiply both sides of the equation by the inverse of r T r, which is a square matrix with dimensions n n. The resulting matrix equation is: B = ( r T r) 1 r T D (5.4) Let a = r T r. a is called the information matrix, and a jk = all t r pj (t) r pk (t). In another version of the linear least squares fit, the r j are normalized by the change r pj (t) in the parameter such that A jk = r pk (t) p j p k. The inverse of this version of the all t information matrix, here written as A, is called the covariance matrix, C. The fourth chapter of reference [6] discusses a similar fit method that uses the covariance matrix. The covariance matrix gives both the correlation between different parameters and the uncertainty in the post-fit computed value of each parameter. The diagonal elements, C jk where j = k, correspond to the square of the uncertainties in each postfit computed parameter p j. The off-diagonal elements, C jk where j k, represent the square of the correlations between parameters p j and p k. The more correlated the two parameters, the more difficult it is to distinguish the effect of one parameter from 62

64 the other on the Earth-Moon distance. Numerical instabilities currently prevent an accurate calculation of the covariance matrix: the terms in A vary greatly in orders of magnitude. The fit code uses the version of the linear least squares fit outlined in this chapter, with a jk = r pj (t) r pk (t), because it avoids those numerical instabilities. all t The determination of the covariance matrix is reserved for future work. One potential solution consists of optimizing the units such that the elements are comparable in size. Let Y = r T D in Equation 5.4. Then, B can be found using Python s function np.linalg.solve(a,y), which solves ab = Y. Similar to Equation 5.2, the post-fit computed parameter value equals the pre-fit computed value p Ci plus the finite change in the parameter times the coefficient b. For each parameter p j, p j,cf = p j,ci + b j p j. This equation keeps with the assumed linear dependence of r j on p j. If the parameter value changes by b j p j, then r changes by b j r j. The post-fit computed parameters can then be used to simulate the final model of the range measurements. The post-fit residuals r O r Cf measure the success of the fit. An unweighted linear least squares fit, this method does not include errors in the measurements. All data points receive equal weight in the fit. 63

65 Figure 5-1: The plots above show r versus time when a parameter is changed by pj, given in Table 5.1. r = r(pj = pj0 + pj ) r(pj = pj0 ). pj was adjusted to produce a maximum r of a meter over the course of the two-year simulation. The masses are in units of kilograms, velocities in meters per second, and position coordinates in meters. 64

66 Chapter 6 Simple Tests of the Fit Some basic tests ensure that the fit is working correctly. These tests probe the limitations of the code when fitting more than one parameter and when fitting parameters that have large differences between their observed and pre-fit computed values. The latter is of particular importance for the parameter η since it has a fractional change of 0.05 but differs by 1 for a Newtonian and full SEP violation model. The post-fit computed parameter values cannot be properly compared to the observed values without the covariance matrix, which provides the uncertainties in the post-fit computed values. In the absence of the covariance matrix, the percent difference between the post-fit computed and observed values serves as a measure of the fit s accuracy. Any difference on the order of a percent or less is regarded as a successful fit. 6.1 Fitting a Single Parameter An initial test of the code consists of fitting a single parameter for which the pre-fit and observed parameter values are different. One such test had an observed η of The data was fitted using a full SEP violation pre-fit model, with η equal to 1. The code gave a post-fit computed η of , different from the observed value, used to generate the data set, by 0.015%. Similarly, for an observed η of 0.9 and a pre-fit computed η of 0.81, the fit derived a post-fit computed η of , 65

67 0.0014% different from the observed value. Both of these tests had a duration of two years and a time step of 1000 seconds. Decreasing the time step to 500 seconds resulted in post-fit computed η values of and for otherwise identical conditions to the tests above. These post-fit computed parameter values are 0.1% and 0.027% different, respectively, from the observed values. While the percent differences would indicate that the fit decreased slightly in accuracy, a true comparison requires the uncertainties σ η in the post-fit η for the simulations with dt = 500 and dt = 1000 seconds. These uncertainties would be obtained from the covariance matrix. The σ η for these two simulations are likely indistinguishable. Unless otherwise noted, a time step of 1000 seconds is used for all fits. 6.2 Fitting Multiple Parameters The test documented in the first section of this chapter may be extended to fit additional parameters. The first of these tests fitted r my in addition to η. The prefit computed η and r my were 0.0 and 1.5 meters, respectively, while the observed values were 0.1 and 0.0 meters. The post-fit computed values were and meters. The post-fit computed η diverges from the observed value by about 0.25%. Repeating this test for m e and η yielded similar results. For a pre-fit computed η of 0.0 and m e of kilograms and observed values of 0.1 and kilograms, respectively, the post-fit computed values were and kilograms. The limits of the code become more apparent when three parameters are allowed to vary in the fit. In one such test, the observed mass of the Earth was kilograms, the observed η was 0.1, and the observed Moon s y-velocity was m/s. The pre-fit computed values consisted of the Newtonian default parameters: an m e of kilograms, η of 0.0, and v my of m/s. The post-fit computed values for these parameters were kilograms for m e, for η, and m/s for v my. While the post-fit computed 66

68 v my falls within % of the observed value, η exceeds the observed value by a factor of 15. The lessened accuracy of the post-fit computed values comes in part from the ability of parameters to absorb the effects of the others, to varying degrees. An increase in the number of parameters being fitted compounds the degeneracy: the fit has more degrees of freedom to minimize the residuals between the model and the observed data sets. However, an examination of the values would suggest that the poor fit occurred largely because the difference of m/s between the pre-fit and observed Moon s y-velocity greatly exceeds m/s, the amount p by which that parameter was perturbed when generating r vmy. The fit method assumes that the Earth-Moon distances depend linearly on the parameters. This assumption remains valid so long as the model and observed values for the parameter differ by about p. For parameters on which the distance depends linearly, the partials, or in this case finite differences, are constant for all values of the parameter. However, the Earth-Moon distance does not depend linearly on the parameters, so the calculated differences only describe the dependence of the distance on the parameter in the vicinity of p and p + p. If the difference between the pre-fit computed and observed v my is reduced to m/s, the fit becomes better, giving values of kilograms for m e and for η, though the post-fit η is still greater than the observed value by a factor of four. When the difference is further reduced to the order of magnitude of v my = m/s by using a pre-fit computed v my of m/s and an observed v my of m/s, the post-fit computed values are v my = m/s, η = , and m e = kilograms. The post-fit computed values of v my and m e match the observed values. The post-fit computed value for η is only 1.6% different from the observed value. 6.3 Fitting All Parameters When η O η Ci These tests consist of fitting all parameters when the pre-fit computed and observed parameters are identical except for η. A very basic precursor to this test involves 67

69 fitting parameters for which identical values, including that of η, were used for both the model and observed data sets. The fitted values should match the observed values, which was the case. Fit 1 has an observed η of 0.1 and a pre-fit computed η of 0.0. Table 6.1 gives the post-fit computed values for all the parameters. Several of these values match the observed values exactly, such as v ey, v my, and m e, while others differ from their observed values by a negligible amount, as is the case for the x-velocities with observed values of zero and post-fit computed values on the order of 10 5 or less. The post-fit computed value of η differs by only 1.3% from the observed value. With a pre-fit computed η of 0.1 and observed η of 0.0, Fit 2 gives very similar results. The post-fit computed value of η is about Fit 3 uses an observed η of 0.0 and pre-fit computed η of 1.0, while Fit 4 uses the reverse. Both resulted in post-fit computed values very near the observed values. Some post-fit computed and observed values were identical. These parameters include v ey, v my, and r mx. Overall, the code fitted the values slightly less accurately, but even so, the inaccuracies remain minuscule. For example, the post-fit computed value of m m differs from the observed value by % for Fit 4. The post-fit computed η for Fit 3 differed from the observed value by 0.055%, and while the η Cf η O is on the order of 10 4 for Fit 4. An observed η that differs from the pre-fit computed η by the largest amount possible, 1, does not render the fit inaccurate in the slightest. The dependence of r on η appears to be linear. The plots of r η for these tests are identical regardless of the value of η at which the finite differences were calculated. Figure 6.3 plots r versus time at η = 1 and η = 0. A difference between the two curves becomes apparent as time increases, so the dependence of r on η is not perfectly linear. However, so long as the duration of the fit does not exceed four years, the non-linearity should have a negligible effect. This finding keeps with earlier results. In Chapter 4, the difference between a quasi-newtonian and Newtonian simulations was found to be proportional to η for a three-year duration. Differencing two simulations with slightly different values of η gives the r used in the fit. Just as r = r(η 0) r(η = 0) 68

70 t Figure 6-1: The finite differences in the Earth-Moon distance computed at η = 1 and η = 0 with η = 0.05 are nearly indistinguishable for the first few years of the five-year long simulation. The differences become noticeable after three years. was proportional to η, r = r(η = η 0 + η) r(η = η 0 ) is proportional to η and independent of η. 69

71 Fit η O η Ci m e m m m s v ex v mx v sx v ey v my v sy r ex r ey r sx r sy r mx r my η Cf Table 6.1: These four tests had the same observed and pre-fit computed parameter values for all parameters but η. The table lists post-fit computed values. The pre-fit, post-fit, and observed eta are each specified. The masses are in units of kilograms, velocities in meters per second, and position coordinates in meters. 70

72 Chapter 7 Fitting an SEP Violation with a Newtonian Model In order to isolate the SEP violation signal, a Newtonian model must be fitted to quasi-newtonian distances, generated with η = 1. The fit does not allow the model η to vary; η remains frozen at zero. Ideally, the post-fit computed parameters will adjust to eliminate the other periodic signals associated with the eccentricity of the lunar orbit and the Sun s perturbation of the Earth-Moon system, leaving only the Nordtvedt Effect in the residuals and Fourier spectrum. Four fits of this kind were run with the same initial conditions with the following exceptions: Fit A had a time step of 1000 seconds and a duration of two years. Fit B had a time step of 1000 seconds and a duration of three years. Fit C and D had durations of two years and time steps of 500 and 100 seconds, respectively. Figure 7-1 presents some initial and promising results of this method. For Fit A, with a duration of two years and a time step of 1000 seconds, the Nordtvedt Effect signal appears to dominate the residuals. The Fourier spectrum reflects the prominence of the SEP violation signal, the only significant signal in the spectrum. The other signals have amplitudes of two meters or less. The Fourier spectrum for the two-year long simulation presented in Chapter 4 indicated that the three of the four main signals were identifiable. The Fourier spectrum of the Fit A post-fit residuals therefore indicates some success in absorbing the other signals. 71

73 The post-fit computed parameters have absorbed some of the difference in the lunar orbit s eccentricity for the SEP violation and Newtonian cases, which would otherwise produce a prominent signal at days. However, the poor resolution likely obscures the portion of the eccentricity signal that the post-fit computed parameters failed to fully absorb. The spectrum exhibits shallow, wide peaks in the vicinity of the SEP violation signal. It remains a possibility that these peaks would morph into more prominent, distinct signals with increasing duration and resolution. Based on the 11.5 to 11.7 meter prediction and the result in Chapter 4 for the direct subtraction method, the amplitude of the SEP violation signal falls about 2 meters short of the expectation, indicating that the post-fit parameters absorbed some of the SEP violation effect. The persistence and strength of the signal, however, indicate that the post-fit computed parameters, which do not include η, barely mask an SEP violation. Table 7.1 lists the post-fit computed parameter values for Fit A, which have changed from their pre-fit computed values to minimize the residuals. The results of a similar fit, B, are shown in Figure 7-2. The duration was increased to three years, and the p j were adjusted to produce a maximum of r = 1 meter over the course of the simulation. The post-fit residuals reverted to the wavepacket structure seen in Chapter 4. The Fourier spectrum exhibited similar signals to those in the spectra of the Newtonian distances directly subtracted from the quasi- Newtonian distances. The post-fit computed parameters did successfully absorb 2 meters of the contribution of the eccentricity to the Earth-Moon distance, and the SEP violation signal once again falls about 2 meters short of the expectation. The reason for the persistence of the other three signals in the post-fit model remains unclear. Either the post-fit parameters failed to absorb those effects, associated with η being different in the model and observed data, or the post-fit parameters introduced additional signals at these periods. For example, the post-fit parameters for the Earth s velocity components, which differ from their observed values, will affect the Sun s perturbation of the Earth-Moon system and the shape of the lunar orbit. The post-fit computed parameters may therefore introduce differences between the post-fit computed distances and the observed distances at periods of approximately 72

74 Figure 7-1: The SEP violation signal dominates the post-fit residuals of Fit A, which has a duration of two years. With a 10 meter signal at the synodic period, the Fourier spectrum indicates that in the absence of a non-zero η, the post-fit parameter values cannot mask an SEP violation. 31.8, 27.55, and days. Decreasing the time step from 1000 seconds to 500 seconds fails to significantly improve the post-fit residuals. Fit C had the same conditions as Fit A with the exception of the time step. Figure 7-3 presents the results. Like Fit A, the SEP violation signal dominates the residuals and has about the same amplitude, 10 meters, as the signal in Fit A. The signal at about 14 days also remains a prominent feature in the Fourier spectrum. Fit D also had the same initial conditions as Fit A with the exception of the time step, which was further reduced to 100 seconds. The quality of the fit deteriorated noticeably: the residuals shown in Figure 7-4 exceed 50 meters. The residuals resemble many of the r plots from Figure 5.1, such as one the corresponding to m e. It seems as if those post-fit computed parameters with r plots resembling the the post-fit residuals overcompensated for the SEP violation. As a result, the post-fit residuals mimicked the offsets in the Earth-Moon distance that these parameters produced when perturbed by p. However, given in Table 7.1, many of the post-fit parameter values for Fit D are closer to the observed values than they were in Fit A. Perhaps the combination of postfit parameters resulted in the unusual shape of the residuals and more pronounced 73

75 Figure 7-2: Using a duration of three years, Fit B has post-fit residuals with an envelope shape. Both the residuals and the Fourier spectrum, with four signals, resemble the results from Chapter 4, in which Newtonian and full SEP violation simulations were differenced. non-sep violation signals in the Fourier spectrum, also shown in Figure 7-4. Another possibility is that the small time step may have rendered the calculations more prone to certain numerical errors, which markedly decreased the quality of the fit. Or, these results may indicate an error in the code that has yet to be resolved. The Fourier spectrum still includes an SEP violation signal of a 10 meter amplitude. 7.1 Fitting Newtonian Observations with a Full SEP Violation Model The following fits assume a full SEP violation model, while a Newtonian framework generated the observed distances. This full SEP violation model with η frozen at 1 should be unable to absorb the 12 meter offset of the Newtonian lunar orbit. It should therefore exhibit the same characteristic signal of the Nordvedt Effect in a Fourier spectrum of the residuals. For a three-year, dt = 1000 second simulation, the SEP violation signal dominates the post-fit residuals in Figure 7-5. The Fourier Spectrum reflects the strength of this signal and the lack of other significant periodic behavior: it has no other peaks with amplitude greater than two meters. However, for 74

76 Figure 7-3: With a time step of 500 seconds, Fit C s results strongly resemble the results of Fit A, which had a time step of 1000 seconds. a two-year long simulation, the post-fit residuals once more exhibit strong signals at periods other than the synodic period as shown in the Fourier spectrum in Figure Implications Each of the presented fits exhibit the SEP violation signal per the Nordtvedt Effect. The expected amplitude of this signal is about 12 meters. In these spectra, the signal often has an amplitude of approximately 10 meters, suggesting that the postfit computed parameters are absorbing about two meters worth of the SEP violation signal. The conditions that resulted in the SEP violation signal dominating the postfit residuals in some cases but not others remain unclear. The quality of the fit seems to be very dependent on the duration of the simulation and the p values. This is neither expected nor yet understood. Additionally, the other peaks in the Fourier spectra persist. Similarly located peaks appear in the Fourier spectrum of the directly subtracted SEP violation and Newtonian data sets presented in Chapter 4. As the post-fit parameters diverge from the observed parameters, they may impact the Sun s interaction with the Earth- Moon system and the shape of the Moon s orbit, thereby introducing signals into the residuals. Or, the post-fit parameters failed to absorb the impact of an SEP 75

77 Figure 7-4: With a duration of two years and a time step of 100 seconds, Fit D produced larger residuals which lacked both a shape dominated by the Nordtvedt Effect and a series of wave envelopes. The Fourier spectrum includes the SEP violation signal of 10 meters at the synodic period. However, it also includes two strong signals the strongest of the fits presented in this chapter around 14 and 27 days. violation on the Sun-Earth-Moon interaction and on the lunar orbit s eccentricity. Even in those fits which appear to be more successful, the limited resolution may be obscuring the remains of the non-nordtvedt Effect signals. Most likely, these fits did not entirely absorb the other signals. The peaks may also have persisted due to numerical errors that worsened depending on the duration and p values used in the fit. To explore this possibility, two fits with different levels of precision were compared. A fit used the np.float128 option to see if the increased precision eliminated the other peaks in the spectrum of the residuals. The initial result indicated that the np.float128 version fitted the data much more poorly, with residuals on the order of 100 meters. After reducing the the changes in the parameters to produce smaller offsets, on the order of a tenth of a meter, in the Earth-Moon distance, the residuals improved by an order of a magnitude, but still failed to match the expected results. Python s linear algebra functions do not support np.float128 and therefore had to use np.float64 even as the initial conditions and simulation used np.float128. The mixed levels of precision may be responsible for these bizarre results. This test of precision is inconclusive, and the accuracy of the simulation and potential sources of the extraneous periodic signals 76

78 Figure 7-5: With a duration of three years and a time step of 1000 seconds, the most successful fit involved fitting Newtonian data with a full SEP violation model for a duration of three years. The only prominent signal in the Fourier spectrum of the residuals is the SEP violation signal. deserve more investigation. 77