Celestial Mechanics III Time and reference frames Orbital elements Calculation of ephemerides Orbit determination
Orbital position versus time: The choice of units Gravitational constant: SI units ([m],[kg],[s]) G = 6.67259 10-11 m 3 kg -1 s -2 Gaussian units ([AU],[M ],[days]) k = 0.01720209895 AU 3/2 M -1/2 days -1 Kepler III: k 2 replaces G m 1 = 1; m 2 = 1/354710 (Earth+Moon) P = 365.2563835 (sidereal year); a = 1
Celestial mechanics units The usage of Gaussian units is typical of celestial mechanics applications In spaceflight applications, SI units are also necessary to use In Oort Cloud dynamics, one often uses the year as unit of time. The gravitational constant is then = 4π 2 to very good approximation
History of time keeping Before 1960: Earth s rotation provided the basic clock, measured by astronomical observations of the sidereal day = 23 h 56 m 4.1 s (Earth s spin period). One mean solar day = 24 h = 86400 s. Universal Time (UT) = Greenwich mean time (GMT). Locally observed UT = UT0 (the results differ due to polar motion). Corrected UT = UT1 (but the rate still varies because the Earth is not a perfect rotator). Lunisolar tides lead to Length Of Day (LOD) variations.
LOD variations From Time Service Dept., US Naval Obs. Note: the LOD differs from 86400 s by about 2 ms!
History of time keeping, ctd From 1960: Solar system orbital motions provided the basic clock, measured by astronomical observations of orbiting objects. Ephemeris time ET = uniform time, as required by the Newtonian equations of motion. The ephemeris second was defined as: 1/31,556,925.9747 of the tropical year at epoch 1900 UT1 was used temporarily, until ET was available and the correction ΔT=ET-UT1 was published.
History of time keeping, ctd In 1967: the frequency of a hyperfine transition in the 133 Ce isotope defines the SI second to be the duration of 9,192,631,770 periods of the radiation corresponding to this transition. This is equivalent to one ephemeris second. International Atomic Time (TAI) is based on the SI second and maintained by a large number of clocks operating at standards laboratories. Zero point: UT1 TAI 0 on Jan. 1, 1958 Since 1984: Terrestrial Time (TT) is used in astrometry and ephemeris calculations, thus replacing ET. Zero point: TT TAI = 32.184 s (ET UT1 in 1958).
History of time keeping, ctd Currently, on average, one mean solar day = 86400 UT seconds 86400.002 TT seconds. In about 500 days, the difference between TT and UT increases by 1 s. Coordinated Universal Time (UTC) increases at the TAI rate like TT, but by introducing leap seconds, it is kept close to UT1. It defines civil time and provides the connection between astronomical and atomic times. The first leap second was introduced in 1972 with a starting value of TAI UTC = 10 s. The latest one (nr. 24) was introduced on 31 Dec. 2008.
Julian dates A continuous count of days and fractions since noon UT on Jan 1, 4713 BCE (Before Christian Era) on the Julian calendar; the day numbers are now approaching 2.5 million Very useful for ephemeris calculations Converters between Julian dates and calendar dates can be found at: http://aa.usno.navy.mil/data/docs/juliandate.html A practical formula, valid for the 20 th and 21 st centuries, is:
Fundamental reference frames Ecliptic frame, basically heliocentric, couples to the orbits of objects Equatorial frame, basically geocentric, couples to the astrometric observations Each one is defined by a fundamental plane, which cuts the celestial sphere along a great circle. The points of intersection are the equinoxes the vernal equinox is used as reference direction.
Reference frames, ctd The angle between the two planes ( obliquity of the ecliptic ) was ε=23 26 20 on 1 Jan. 2000. Due to lunisolar precession of the Earth s spin axis, the vernal equinox drifts by ~50 per year. Planetary precession of the ecliptic plane causes smaller effects in both ε and the equinox. Since we need fixed reference frames, we use a standard epoch (currently J2000.0 ) to define the equator and equinox.
Angular coordinates Use (x eq,y eq,z eq ) and (x ec,y ec,z ec ) as Cartesian geocentric coordinates. Let the x eq and x ec axes point toward the vernal equinox, and let the z eq and z ec axes point toward the respective poles. In spherical geocentric coordinates we instead use: Δ = geocentric distance α = right ascension δ = declination λ = ecliptic longitude β = ecliptic latitude (index 2000 means they refer to the standard equator and equinox of 2000)
Coordinate transformation
Orientation of orbit w.r.t. ecliptic
Inclination i is the angle from the pole of the ecliptic to the pole of the orbital plane (0 < i < π) Prograde orbits: i < π/2 Retrograde orbits: i > π/2
Longitude of the ascending node Ascending node vector Ω is measured along the ecliptic, counterclockwise as seen from the North pole, from the vernal equinox to the ascending node
Argument of perihelion ω is counted along the orbital plane, counterclockwise as seen from its North pole, from the ascending node to the perihelion direction
Orbital elements We have identified six orbital elements, which can be grouped as follows: Semi-major axis a, or perihelion distance q = a(1 e) Eccentricity e Time of perihelion passage T, or mean anomaly at a given epoch M 0 Orbital position at given time Inclination i Longitude of the ascending node Ω Argument of perihelion ω, or longitude of perihelion ϖ = Ω+ω Conversion from orbital position to ecliptic frame
Conversion matrix This transforms vectors from the orbital frame to the ecliptic frame, e.g., the position and velocity vectors:
Ephemeris calculation Calculate mean motion n=2π/p, P=2πa 3/2 /k Calculate mean anomaly M=n(t-T) Solve Kepler s equation to obtain eccentric anomaly E Use E to calculate the position (and velocity) vector(s) X (and dx/dt) in the orbital frame Use {i,ω,ω} and Homer s transformation matrix to obtain position (and velocity) in the heliocentric ecliptic frame
Ephemeris calculation, ctd Find the heliocentric ecliptic coordinates of the Earth Calculate the geocentric position vector of the object Calculate the distance Δ of the object Correct for planetary aberration (light time correction) If the light time is Λ = Δ/c, repeat the calculation of the object for a time t Λ
Ephemeris calculation, ctd Convert to equatorial coordinates Find the right ascension and declination of the object Repeat for a set of regularly spaced dates
Orbit determination Essential in order to identify and keep track of moving objects like Near-Earth asteroids or comets Clearing house: IAU Minor Planet Center Search programs, discovery statistics for numbered minor planets as of March 10, 2009: LINEAR 104,780 Spacewatch 22,036 NEAT 19,670 LONEOS 13,000
Way of procedure Preliminary orbit determination according to one of several methods (calculate orbital elements from few observed positions) Orbit improvement (reduction of uncertainties in orbital elements by using many observed positions) Linkage of several oppositions of an asteroid or apparitions of a comet (including chance identifications)
The Method of Gauss Assume 3 observations available: (α j,δ j,t j ); j=1,3 Thus 3 geocentric equatorial unit vectors are known: We can also find the Sun s geocentric positions at the 3 times of observation We can write 3 heliocentric position vectors of the object using unknown geocentric distances: (at time t j )
The Method of Gauss, ctd Idea: find the three Δ values; then use the three r values to derive the heliocentric position and velocity at the middle observation Transform from position and velocity to orbital elements Since motion is planar: Coefficients given by:
The Method of Gauss, ctd We obtain a vector equation, equivalent to 3 scalar equations, from which Δ j can be solved, if c 1 and c 3 are known: The method will be to work with successive approximations and iterate until convergence Start with an initial guess for c 1 and c 3
Estimating the c parameters Half the vector product is the area of the triangle; A is the area overswept by r y j are the sector-to-triangle ratios (close to 1) by Kepler II: Similarly:
Transforming the vector equation Transform from the equatorial to a new coordinate system, which we denote C : Multiplying the vector equation for Δ j by the transformation matrix, in the new system we have:
Solving for the Δ s Due to our choice of axes in the C system, the scalar equations become separable in the three Δ s The scalar products ξ i, η i,ζ i are known quantities
Solving for the Δ s, ctd
Algorithm Use {α j } to calculate Δ j unit direction vectors Use these to calculate {ξ 2, η 2, ζ 2, ξ 3, η 3 } and R eq C Transform Δ j and R,j vectors to C system We need to know c 1 and c 3! Assume y 2 / y 1 = y 2 / y 3 =1 Calculate the geocentric distances! Calculate the approximate heliocentric position vectors of the object! But now we can estimate overswept areas, i.e., more realistic values of c 1 and c 3 can be calculated. Iterate!
Numerical example
Orbit Improvement Apply an algorithm to the orbital elements to obtain an ephemeris Π: Orbital elements Θ: Algorithm (e.g., two-body) Φ: Ephemerides Compute the Jacobian of the algorithm: Further observations show discrepancies with the ephemeris
Orbit Improvement, ctd Identify the Jacobian with the ratio of finite differences: With many observations, we get a system of many equations involving known residuals δφ. Use a statistical method like the least-squares method to find the δπ that minimizes the residuals.