Jordan McDonnell. Physics and Astronomy Department. Effects of a Modified Gravitation Law on the Perihelion Shift. 12 May 2007

Transcription

1 Jordan McDonnell Physics and Astronomy Department Effects of a Modified Gravitation Law on the Perihelion Shift 12 May December 2006

2 Table of Contents Abstract 1. Introduction 2. Review of Classical Orbital Mechanics 3. Special Case - Small Eccentricity Orbits 4. Generalized Precession Formula 5. Perihelion Shift - Power Law Force with Any Eccentricity 6. Numerical Results 7. Applications 8. Conclusions 9. Bibliography

3 Abstract The perihelion shift in a satellite's orbit has become an important measure of the effects of non- Newtonian contributions to the force of gravity. Historically important contributions have been ascribed to general relativity, and more recently there has been interest in using measurements of the perihelion to corntrain parameters for forces from darlc energy, as well as explaining anomalous motion of artificial satellites. The functional form of the perihelion shut seen in the cases of general relativity and dark energy, calculated in the literature, is seen in the present work to generalize to other forces. The present work hds a generalized formula for the precession caused by any central force for small eccentricity, as well as an integral giving the precession for any force with any eccentricity that b not been encountered in previous literature. In the case of power law forces, this integral is performed analytically. These results are analyzed numerically, and the new formulas are applied to recent applications in dark energy, anomalous satellite motion, and gravitational contributions from higher-dimensional theories.

4 1. INTRODUCTION High-precision measurements of astronomical motion have been of interest in obtaining increasingly refined descriptions of gravity since Kepler fit ellipses, rather than circles, to planetary orbits. Newton's theory of universal gravitation provided an explanation for Ke- pier's ellipses and gave detailed predictions for the positions of each planet that agreed well with observations. When the planet Uranus was found to deviate slightly from the New- tonian predictions, Couch and Leverrier independently located the position of an unknown body that could be exerting the gravitational influence on Uranus that would account for its deviant motion. When astronomers investigated that position, they discovered the planet Neptune. The success of Newtonian gravity in providing not only a summary of observa- tions, but also the power to predict forces due to presently unknown sources, was considered a great triumph of the theory [I]. The planet Mercury, however, was found to deviate slightly from Newtonian predictions (by experiencing a shift in its perihelion, the point at which the planet is closest to the sun, of 43" per century in excess of that produced by known effects). Several explanations were offered, ranging from Newcomb's suggestion of a gravitation law with a power of r not exactly equalling -2 to a hypothetical collection of objects (possibly planets) between Mercury and the Sun. Einstein's theory of general relativity ultimately predicted a shift in Mercury's perihelion of precisely the 43" per century needed. This prediction, along with several other successful tests such as light curving around the sun, led to the widespread acceptance of general relativity [2], [3]. With the scientific description of gravity becoming more refined and better understood through such tests, modern-day problems in gravity require increasing precision to differ- entiate the predictions of competing theories. An important class of measurements is the secular motions of a body, which accumulate wit11 each orbit and can be measured with great accuracy. One such secular motion is the shift of a satellite's perihelion (taken to be the satellite's point of closest approach to whatever body it orbits, whether that body is the sun or not)? as has been historically important in the conhmation of ~eneral relati~ty. Measurements of the perihelion shift have been put forward in several recent cases as a possible means of bounding gravitational effects not accounted for by general relativity as it stands. The question has been raised as to whether the perihelion shift might provide

5 information about dark energy by placing an upper bound on the cosmological constant 141. The anomalous motion of the Pioneer satellites has also prompted investigations into whether there exist significant deviations from Newtonian mechanics or additional, unknown forces [5] -such forces would presumably affect the motions of the planets in the solar system, possibly through a measurable perihelion shift. Further, higher-dimensional theories (variations of string theory) have been seen to modify gravity on some scale, producing some kind of effect on planetary motion [GI. In the theoretical investigation of such apparently anomalous motion, some small force in addition to gravity is introduced by some means (most often as a new source of potential energy). General relativity, for example, treats dynamics in terms of the metric describing the curvature of spacetime, and the trajectories of satellites are described by geodesics. The effects of dark energy and the cosmological constant can also be accommodated within the metric approach of general relativity. Analyses of several other effects (such as the Pioneer anomaly and higher-dimensional theories) also typically begin with a variation on a basic relativistic metric. Because these additional corrections are always small, some approximation such as perturbation theory has been appropriate for all cases of practical interest. In the process of approximation, the equations of motion have typically reduced to the familiar Newtonian equations of motion with some force added to gravity. The present work uses this observation and begins with Newton's equations of motion to derive several expressions for the perihelion with various ranges of applicability. In the literature, a variety of approaches have been applied to obtain the precession of a satellite's perihelion. In so doing, the mathematical form of an arbitrary correction has not yet been put forward. The work of Ohanian [2] found an expression for the precession caused by general relativity (assuming nearly circular orbits), but this approach is generalized in the present work to other forces. Furthermore, work done by Kerr, Hauck, and Mashhoon 171, Rindler 181, and Wright [9] have given expressions for the precession caused by dark energy; the expressions, however, are in disagreement. Fkom mere inspection, it is not straightforward to determine which expression is correct, so it is necessary to attempt a new derivation from Erst principles. It has become desirable to obtain a formalism that can accommodate all possible corrections to gravity, including those of present phenomenological interest.

6 After reviewing orbital mechanics in general (loosely following Goldstein [lo]), the present work will first follow the approach of Ohanian [Z] and assume nearly-circular orbits for small corrections to gravity, and calculate the precession that results due to any form of the force. Secondly, a new approach will be employed to derive the precession for any possible eccentricity, using approximations dependent on the smallness of the additional force. This approach results in an integral that can always be evaluated numerically, and it will be seen in the case of a power law additional force to be analytically integrable. In the case of power law forces, the analytical predictions will be tabulated and compared with purely numerical results. Finally, the applications (namely, dark energy, the Pioneer anomaly, and a possible gravitational correction arising from string theory) are discussed, comparing measurements done by Pitjeva [ll] on precessions of planets in excess of those caused by known effects to the precessions calculated from the present work's results. 2. REVIEW OF CLASSICAL ORBITAL MECHANICS The discussion of orbital mechanics will concentrate on a satellite orbiting some body under the influence of only a radial force (namely, directed towards the body). The body is assumed to be much more massive than the satellite, so that its own motion proves negligible. The motion of the satellite is determined by Newton's second law, where F is the vector pointing from the body to the satellite (with f the unit vector in that direction), and f (r) is the (general) central force; p = mg is the satellite's momentum. One quantity to consider is the angular momentum mh = f x 5 (h is here defined to be the angular momentum per satellite mass), which is seen to change in time as the cancellation at the second step utilized the fact that cis a scalar multiple of $, and the third step employed Newton's law for a central force. This indicates that h is a conserved

7 quantity. In particular, f and 5 must always lie in a plane normal to h, and so the entire motion takes place in a plane. Taking this plane to be the xy plane, the components of T may be written 7 = (r cos if)); + (r sin if))j, where r remains the magnitude of the radius vector and ip is the azimuthal angle. It is then seen that and d' dt - = (f cos if) - r4 sin $); + (?sin 4 + r4 cos if)); d2? - = [(f- T@) cos if) - (2f(* + r<f>) sin if)]; + [(r- r@) sin ip + (2f4 + r4) cos if)}], dt2 (4) where the overdots denote differentiation with respect to time. Using the unit vectors for? = cos 4; +sin ipj and = -sin <bi + cos 6, these expressions simplify considerably by taking dot products to obtain the components in polar coordinates: (3) and The second of these may be substituted into (I), since $f = mg, to obtain the two eauations The second of these equations is parallel to the previous statement of conservation of angular momentum, which is seen by multiplying both sides by r and taking an indefinite integral to obtain a constant h, Indeed, from the above equations for g, it is seen that

8 so that (7) is identically the magnitude of angular momentum per satellite mass, h. Solving for (f> from (7) and substituting the result into (5) produces This equation may be multiplied by f and integrated to yield another constant of the motion, 1. h2 -r V(r) = &, 2 2r2 where V(r) = - f f(r)dr is the potential energy per mass, and the & is a measure of the total energy per satellite mass. Following the standard approach of Goldstein [lo], variables may be separated to obtain (9) (In taking the square root in the solution, the negative sign has been chosen to accommodate a later step in the calculation. The choice of sign here determines whether the satellite encounters fust its maximum or its minimum distance from the body.) This equation may clearly be integrated, at least numerically, for any potential function V(T). In special cases, the integral may be performed analytically, as discussed by Goldstein [lo]. The case where the force law is proportional to r 2, however, is special in that this integral may be performed directly with a trigonometric substitution. Take the force law to be f(r) = -?, so that the potential is V{r) = -?, where G is Newton's gravitational constant and M is the mass of the body about which the satellite orbits. The equation (10) becomes which may be manipulated by two key transformations. First, from (7) the dt can be transformed to d$ by Second, it proves useful to cliange variables on the right hand side to u = 3- (and dr = -$du). These transformations yield

9 which may be simplified (and integrated) to yield P where (do is some constant related to initial conditions (and is often arranged to be 0). The argument of the square root function may be rearranged by completing the square: It is now possible to introduce a dummy integration variable a, dehed by where C = GM -F = Csin a; du = C cos ada, 1 + IGM}'-. The integral in (11) is seen to reduce to GM/2Â k (d-(do=jd Ccosa sin2 a da= Jito=a=sin-l( c2 - c2 u-- GM ch'), where the Pythagorean identity has been employed, and the final step follows from the definition of a. This may finally be solved for u (or r) as a function of 4: This is geometrically the equation of a cofic section with eccentricity c = 4- and semilatus rectum L, given by % = F, with one of the foci at the origin (see Fig. 1). As usual, the eccentricity determines which conic section is actually traced out: a hyperbola for e > 1, a parabola for e = 1, an ellipse for 0 < e < 1, and a circle for e = 0. Physically, the eccentricity is effectively determined by the energy parameter &, so that a hyperbola 1 GM 2 corresponds to & > 0, a parabola to & = 0, an ellipse to -5 (T) 1 GM to & = -5 (-lã < Â < 0 and a circle. Since the present work is interested in bound orbits (those in which the satellite never escapes to an infinite distance from the body), such orbits must correspond to either ellipses or circles, so that & is taken to be less than 0 for the remainder of the work. It is to be noticed from this solution that the extreme values of u (one of which would correspond to the perihelion) are determined by

10 FIG. 1: The "anatomy" of an ellipse, with the semilatus rectum L and semimajor axis a marked, as well as the focal point. The magnitude of the eccentricity can be intuitively defined using the relationship between L and a: L = a(l - e ). The extrema may also be found from the energy equation (9), with the gravitational potential substituted. Namely, the extrema correspond to the stationary values of r, which occur for? = 0. Once the change of variables u = is performed, this equation takes the form h2 -u~-gmu-e=o, 2 which (when multiplied by 2 and divided through by h2) is seen to be the argument of the square root function in the integral for 4 in (11). Since this equation is simply a quadratic, the quadratic formula may be applied to find the extreme values of u. This solution yields, under appropriate substitutions for L and e, exactly the roots given in (13). That the energy equation determines the extrema and is further encountered in more generalized problems proves a useful observation. It proves interesting to find the angular change in 4' incurred when u varies from one extreme value to another. This may be determined directly from the solution (12) to be exactly TT. Allowing the satellite to return to its original extremum (that is, it completes an orbit), it is clear that the total change in 4' will be 27r. In other words, under solely gravitational influence, a satellite's orbit is completely closed, returning to the same distance from the body at the same angular displacement each circuit.

11 PIG. 2: A diagram of the shift of the perihelion in the plane of orbit, showing the angle (A in the diagram) between points at which the satellite returns to its greatest distance from the body. The diagram is from R'idler [8], Fig. 11.5, pg It is also useful to directly integrate (11) over the interval sparming the extreme values of u. Factoring the argument of the square root according to the energy equation's zeroes and evaluating the integral yields as has been argued from the solution for ~(4). This particular calculation, utilizing the factoring of the integrand, proves to be a useful model for calculations in a later section. Furthermore, the angular displacement through the whole orbit is seen to be twice the value of this integral: 4 = 2 1 du = 27r. Again, that this result is exactly 27r indicates that the orbit is closed, and calculations of this sort will be useful to quantify the precession of an unclosed orbit in this integral's difference from 27r. A solution allowing closed, perfectly repeating orbits turns out to be very special in the case of gravitation (and other forces proportional to r2). If the angular displacement upon a completed orbit were not 27r, the angular timing of the perihelion would be seen to change

12 with each orbit, so that the nearly-elliptical shape would precess in the plane of the orbit (see Fig. 2). The perihelion shift caused by small perturbations to gravity is to be calculated in the following sections. 3. SPECIAL CASE - SMALL ECCENTRICITY ORBITS The general form for the effects of an additional small force on the perihelion shift can be found from the solution of Newton's second law, beginning with where m is the satellite mass, A6 is the body mass, G is Newton's gravitational constant, and F(T) is the additional (central) force. This equation is divided by m, and F(T) = F(r)/m is the new function equal to the additional force per satellite mass (namely, some extra acceleration). Since the forces in addition to gravity are still central, the analysis of the previom section still holds - particularly useful is the conclmion that angular momentum + (mh) is conserved, which implies that the motion occurs in a plane. Adapting (8) to the current force under question yields This result may be multiplied by f and integrated as before to yield the energy constant: The function U(T) = - f F(T)~T is effectively the potential energy per unit mass due to the additional force. A change of variables u = 1 /~ will prove helpful, as will transforming time derivatives in (16) into derivatives with respect to 4, in accordance with (7): h (u ) + -h u - GMu + U(l/u) = &, 2 2 (17) where primes denote differentiation with respect to 4. Differentiating this equation with respect to (j) and dividing by h2u results in

13 It proves useful to retain the functional dependence of T on r, rather than incorporate the transformed variable u in that portion of the equation. Following the approach of Ohaman [2], this equation is solved by perturbing from the classical solution (12). That is, assume the solution to be of the form where ua is the classical solution and up is the particular solution. Take where A is an unknown constant and e is the eccentricity (which is presently assumed to be small). Since the perturbation is assumed to be small, substitute the classical solution into the last term in (18): 1 1 WA~(I + e sin 4 )~ ~u2^ The assumption of small e allowed several power series to be taken with respect to that variable (near e = O), wit11 terms of order e2 and higher neglected; the multiplication resulting in the last line similarly assumed that e2 is negligible. The second term in the last line may be recognized to be a derivative of a product, allowing the further simplification This is substituted into (It?), along wit11 the trial solution u = uy + up: The cancellation of the first two terms is clear. The following terms, in brackets, may be seen from the classical solution, along with the assumption that T(r) is always small, to simply determine A as $$.

14 The particular solution up = Bed cos 4 is attempted, where B is an unknown constant. Substitution of this trial into the last line of (20) yields The full solution is then Because the additional force F(r) is very small, this determines that $ is small. This ini~lies that which may be substituted into the full solution: where a trigonometric sum identity has been employed. The angular period of this expression is seen to be SO that the difference between this period and 27r is seen to be upon substitution for the value of B. (The (11) in the superscript of A$('') is chosen to denote the difference between the whole angular displacement over an orbit, which is Ad, and the "expected" displacement 27r. It anticipates the idea that this difference would arise from a second term in a series expansion, as is seen in a later section.) One specialized class of forces to consider is that where the force is proportional to some power of T, as in F(r) = krn. The expression then reduces to

15 Using the knowledge from orbital mechanics that A = 9 = rectum of the orbit, and h2 = GML, this expression becomes $, where L is the semilatus This expression gives the precession of the orbit due to a small additional force given by a power law for any integer n, under the assumption that the eccentricity is small. (22) Comment on the Relationship to General Relativity The work of Ohanian [2] found the precession due to general relativistic effects only. That is, Ohanian began from the metric theory of gravity given by general relativity and manipulated the equations to find an equation resembling Newton's law for gravity, coupled with an additional force from general relativity. General relativity determines the motion of a (massive) test particle according to what is called the geodesic equation, where x is the four-vector, with components (t, T, 8,4) in spherical coordinates with light- speed c set to one, of the particle's position; A is a parameter proportional to the particle's proper time; T is called the affine connection. The affme connection is a function that formally accepts two vector arguments and returns a vector value (geometrically, this func- tion is related to derivatives in curved spaces and spacetime). This equation contains two derivatives of a position vector, and that derivative has the traits of a time derivative -this structurally resembles Newton's second law, such as in (1). The function I? is determined from the metric (given by the line element), which is taken to be the Schwarzschild metric in spherical coordinates where as above the M is the mass of some body at the origin (that the test particle in fact orbits); r is the proper time of the test particle. A full analysis of the geodesic equations finds constants of the motion similar to the classical case, namely conserved angular momentum (indicating planar motion, which is effected by setting 8 = 712, and a conserved energy.

16 Dividing through the line element equation by dr2 and substitution of the constants (formally 2GM given by Ohanian), along with multiplication by 1 - Ã a- yields where B is the energy-like constant. The equation bears some resemblance to the energy equations of above (9), with additional terms. The assumption that all motion is slow relative to the speed of light allows the derivative of r to be changed to one with respect to t. Taking a second derivative with respect to t and dividing by 2f yields?=-- GM h2 3GMh2 +-- r2 r3 r4 ' so that the additional force per mass can be seen readily (on comparison with earlier equa- tions such as (15)) to be a power law with n = -4 and k = -3GMh2 = -3(GMl2L. Substituting these quantities into the expression (22) for the precession yields which agrees exactly with the results, for example, of both Ohanian [Z] and Weinberg [3]. This quantity was historically most important in the case of the planet Mercury, whose extraneous precession of 43" of arc per century was not acceptably accounted for until this relativistic correction was put forward. This test ws one of the strongest early codrmations of general relativity, holding promise that further analysis of anomalous precessions would yield more detailed information about non-newtonian forces at work. The expression (21) holds whenever the eccentricity of a satellite's orbit is small, and is more general than the result of Ohanian in that the present expression can account for any arbitrary force law (and the expression (22) can account for any force law proportional to some power of r). It proves possible to further generalize this result by removing the assumption of small eccentricity. The following sections, utilizing a different method, demonstrate this generalization. 4. GENERALIZED PRECESSION FORMULA To find an expression for the perihelion shift that does not rely on the assumption of small eccentricity, it is useful to return to the energy equation (16); this approach can be

17 seen to parallel that of Weinberg [3], in which a similar integral was found directly from an analysis of the general relativistic equations of motion. The variables can be separated as in classical orbital mechanics to solve for the differentials: 1 dt = - /2Â ++-$-2~(r dr. (23) Following further the classical solution, the variable changes u = 1 and dt = A d4 may be employed, and the equation integrated (and doubled) over the interval between the roots of the radicand, to yield here, the u+ are the extrema (often found numerically) of the perturbed orbit. In the case of no additional force (U = O), this is seen to properly reduce to the 27r of a regular orbit. In particular, the integrand may be re-written as a factored form in terms of the classical extrema of the orbit, found as before in (13) to be where L is the semilatus rectum and e the eccentricity of the elliptical orbit. The integral is then 1 $a+ - u)(u - a.) - $U(U-1) It is clear that this expression may be integrated numerically to obtain the total change in angle as the satellite traverses an entire orbit, and the difference between this result and 27~ is the perihelion shift. To obtain a closed form for the total perihelion shift in the orbit, it is tempting to expand the integrand in powers of the additional force U around U = 0. This approach, however, is plagued by artificial infinities. In particular, the second order term contains a factor of the form du. (25) which yields infinities before reaching the limits of integration. Another expansion is neces- sary to cancel these unwanted zeroes in the denominator.

18 Just as in the case where U = 0 the integral's form is made more transparent by factoring the argument of the square root in the integrand, it proves useful to proceed in an analogous fashion. Set N(u) = -s(ul) as an abbreviation, and consider a factoring of the radicand in the form (a+ - u)(u - a-) + N(u) = (u+ - u)(u - u-)(l + S(u)), (26) where uâ are the extrema of the perturbed (U # 0) orbit and Q[u} is an unknown function to be determined. First, the perturbed extrema are found by assuming a form where SÂ are some small parameters (small enough for nonlinear combinations, as well as products with Q(u) to be negligible). After substitution into the left hand side of (26) set equal to zero, it is seen that The right hand side of (26) becomes S* PS It ^(aâ± a+ - a_ after linearization in the 5*. The factoring equation (26) may now be solved for Q', Because G(u) depends on N(u), which in turn is proportional to the perturbing force U(ul), it is seen that Q(u) is of the order of a perturbation It is now straightforward to write (25) in terms of this new factorization as where the binomial approximation has been applied to the factor containing B(u). It is clear from this form that the entirety of the perihelion shift is contained in the second term in this sum, which is denoted by A#^.

19 Because the actual limits of integration u+ are near the classical limits a+, it is possible to find the perihelion shift by integrating over the classical limits. The integral in the final step of (28) may be rewritten by dividing up the interval of integration (calling the integrand J(u)): Because the integrand depends on the small factor ff(u), the integrals over intervals of length 6+ yield a negligible contribution. The integral is then left as With 6+ being small perturbations, binomial expansions may be taken that retain terms linear in <5± But these terms will be of the form so that these products of Q(u) with 5+ are negligible under the present approximation. It is then possible to calculate the perihelion shift with an approximate expression that relies only on the perturbing potential and the classical extrema. This approximation allows the precession per orbit to be calculated for any eccentricity and with any (small) potential. Namely, with Q(u) found as described. One case of interest in which this integral may be performed completely analytically is that of a power law force. This specialization is investigated in the following section. 5. PERIHELION SHIFT - POWER LAW FORCE WITH ANY ECCENTRICITY In order to evaluate the integral contained in A#'^, the functional form of Q(u) must be simplified for the case of a power law force. That is, F(ul) = ku-", and U(u-l) = -- utn+'). It is to be noted that, because U(ul) is assumed to be of a power law form, n+1

20 the following derivation is invalid in the case n = -1 (where the potential takes a logarithmic form). The function Q(u) for a power law takes the form, by substitution of the potential into the definition of N(u) and into (27), Terms have been grouped by powers of u in anticipation of upcoming steps. The integral (29) has the limits but this is seen to simplify to the interval (-1,l) under the change of variables u -+ x defined by J. u= -(!+ex). L This transformation causes Q(u) to take the form Define m = -(n + 1) as an abbreviation. Each of the terms within the brackets may be expanded in a Taylor series. The terms become: and

21 where the coefficients are the binomial coefficients, (naturally, this becomes zero at a finite value of a if m is a positive integer). The first expression in this list was judiciously divided into its even and odd powers of x. From these series, the factor Six) may be written as where terms from the previous expression for G(x) have been combined utilizing the odd and even terms of the Taylor expansions. It is to be noticed that the z = 0 component of each summation is just zero, so it is possible to start the summations at i = 1, as will be done hereafter. Divide the factor in the denominator dependent on x into the terms of the sum: The change of variables effected in Q(x) must be performed explicitly for the actual integral (29): where the definition of the roots a+ has been employed, as well as the definition of x. Substitution of the form of Six) found in (31), as well as the division utilized, yields where quantities independent of' x have beenremoved from tile integrand, and the order of summation over j and integration over x has formally been switched. The second bracketed quantity in (31), which contains all the odd powers of x, combines with the even denominator and integrates to zero.

22 The innermost integral over x may be evaluated with the substitution x = sina, which reduces the integral to a form found in common tables [12]: The resulting summation over j is then evaluated with identities involving the Gamma function T(i) found in 1131: Finally, the summation over i is evaluated by using the definition of a hypergeometric function found in [14]: The function of the form 2F1(p, q; u; y) is the Gaussian hypergeometric function. This function, according to Luke [l5] is formally the solution to the differential equation which in principle could be solved by a series and placed in the form stated above. It also has the property that, if either of the arguments p or q are equal to the negative of any whole number, then the function 2F~(p, q; u; y) '"'ill be a polynomial wit11 degree equal to that whole number. The result of the calculation in (32) is then seen to be (restoring m = -(n + 1)) or solving for h2 = GML from the root equations This function A<^'^ is plotted against the eccentricity for several values of n (for k =.01) in Fig. 3 and Fig. 4. Comparing this expression with (22), this general expression contains a coefficient bearing exactly the form of the small-eccentricity precession. Furthermore, the general expression

23 FIG, 3: A plot of A@-^ against e for several values of n < -1. The steepest function is n = -6, and the shallowest is n = -3. FIG. 4: A plot of ~p against e for several values of n > -1. The largest values occur for n = 3, and the smallest for n = 0. contains the sought-after eccentricity dependence, taking the form of a hypergeometric function of the eccentricity squared along with parametric dependences on the power of r in the force law. It is interesting to rearrange this derivation to find a relationship between the precessions caused by forces proportional to powers of T of opposite sign. Returning to (29), a change of variables u = may be effected to yield

24 Applying a similar transformation to that used earlier, Ti = i(l + ex), transforms the numerator of the transformed integrand to Following steps identical to those applied before, the integral is then which is seen to relate closely to (32). In particular, the double summation terms are equal in the correspondence -(n + 1) Ã n + 2, or n ~t -(n+ 3). If A&') shift due to a force given by an n power law, the correspondence is given by denotes the perihelion This is seen in Luke 1151 to reflect one of the recursion relations between the hypergeometric functions zfi, which relates Upon proper substitutions, this can be shown to be the relationship obeyed. It is interesting to see that the relationship (36) connects the case n = -4 to the case n = 1, which means that the general relativistic correction is related to the correction due to any linear force (such as dark energy). In particular, the relationship may be immediately employed to obtain a general expression for the perihelion shift for n = 1, which yields This result is in agreement with that of Kerr Hauck, and Mashhoon [7] given for the effects of dark energy, which gives a value for k = $2, where A is the cosmological constant and c the speed of light. The present result, however, is in conflict with that of Wright [9], for example, where no dependence on eccentricity was found, and the coefficient is different by a factor of two. As in [7], the present result is direct conflict with %ndler [8] in terms of the dependence on the eccentricity. That is, when the result of Rindler is recast in terms of the orbit's sdlatus rectum, the eccentricity dependence is cancelled. The approach of

25 Rindler is interesting in that it begins with a modified version of the Schwarzschiid metric briefly discussed above. It is to be noted that (33) does not hold for the case n = -1. This force law would produce a potential U(r) oc lnr, which cannot be handled in the derivation above (which assumes the potential to take a power law form as well as the force). This force law can be handled in principle through the application of (29). The precession formula (33), specialized for forces proportional to some power of r, provides a unified expression to approximate the precession due to a great variety of possible perturbations to gravity. Before turning to some of these applications, the validity of the formula is to be tested against direct numerical integration. 6. NUMERICAL RESULTS Because of the approximations used to arrive at (33), it is crucial to verify the result by some independent means. That the expression correctly reduces to that in (22) for small eccentricity offers some support, but a more exact check is required before (33) is expected to provide useful predictions to tet against experiments. As such, the integral (25) is evaluated numerically for a variety of values of k, e, and for several different example powers n. The perihelion shift per orbit is then found by subtracting 27r from the result. The primary concern with evaluating the perihelion shift is the validity of the approximation. In particular, the dependence of the result (33) on the force proportionality is seen to be linear, which is a relic of the approximation technique employed that might not reflect the true behavior of the system. This dependence is tested for several values of e, with some typical results shown in the table below (namely, e =.3 was chosen for presentation). The semilatus rectum L, related to the semimajor axis (chosen to equal I), is set to L = 1 - e2. The product of Newton's constant and the body mass is set to GM = 47r2. Within each table, the numerical results are compared with those of the expression (33). Digits are given until a discrepancy between the theory and the numerical integration becomes visible. It is to be observed from Table I that the values predicted by (33) are well in line wit11 those calculated by the numerical integration for the given sample of integer values for n. The results agree once an error of order k2 is taken into account for each of the analytical results; this arises from the neglect of terms higher than linear k in the calculation. It is to be

26 TABLE I: Comparison of numerical results (in the "calc." rows) with analytical results (in the 'A&'" rows) from (33) for the precession per orbit for several integer values of n at c. = 0.3. The dependence of the precession on k is seen to be nearly linear within the chosen interval, as predicted by the approximation in the preceding sections. If the numerical integration gave a dependence on k that was clearly non-linear, the approximation scheme would be invalid. calc: A@ calc: A& calc: A&' calc: A&' ,00385.DO x lo-' Dl91,00874, OD ,00876,005253, OD x lo-' x lo-' -1 x lo-' -4 x lo-' OD x lo-' 0-2 x 10-~ 0.DO ,00455, calc: ,002283, ~~~~~~~ A$? , Dl86, x lo-',001519, ,0155 calc: ,00311,00625, ~ A&) , , x lo-' -1 lo-t.00304, calc: \ x lo-'\.00407\.00817\, \ 0-1 x lo-' noted that the discrepancy in digits between analytical and numerical calculation is beyond the allowed significant figures due to this order k2 error. That is, for k =.l, only the first two digits after the decimal point (up to the lo2 place) may be trusted; for k =.01, the first four digits may be trusted. The tabulated results indicate that the analytical approximation and the numerical integration agree up to those allowed places. That the disagreement is small, and that the dominant contribution to the precession is close to the linear results predicted by (33), indicate that the approximations used to arrive at (33) are indeed valid for small k. An analysis of the numerical integration methods that

27 FIG. 5: The graph presents the analytical and numerical results for the n = -4 case at e = 0.3. The nearly linear dependence is clear from the graph, validating the approximation scheme employed. Other values of n and E produce similar graphs. manipulates the working precision indicates that the uncertainty is of the order of 10W6 for a typical calculation, indicating that all quantities smaller than that are properly 0. This precision is also better than that of the analytical results, so the comparison between the two is always valid within the precision obtained from the analytical approximation. For the case n = -4, both the analytical and numerical results are graphed in Fig. 5. Similar results can be seen in Table 11, in which fractional values of n have been used, as well as in Table I11 which employs irrational n values. From this sampling of numerical tests, (33) is expected to remain valid for all n # -1. The case n = -1, which would correspond to a potential energy U(r) oc Inr, must be handled by some other method. The numerical integration would remain valid, as would the factoring procedure employed in the present derivation for the general additional force [as in (29)]. Having bolstered the validity of the expression (33) with an independent numerical check, it is useful to apply this result to a few recent measurements of solar system motion that have been put forward in order to place constraints on cosmological parameters, such as dark energy density (or the cosmological constant). A few phenomenologically interesting cases sith fmce laws not proportional. to a. paw of r tail also he investigated, in which (29) must be utilized.

28 TABLE 11: Comparison of precession results for fractional n, with e = 0.3. These computations similarly indicate good agreement between the numerical integral and the approximation. The designation "as 0" indicates a value of the order of 10' or smaller. ~~~~~~~~~~ A+:;.\ , calc: as , ~$ o calc: a , calc: = I,01931 A&) ~6~ ~ , ~,02701 calc: asO,00269,00539, TABLE 111: Comparison of precession results for irrational n, with f = 0.3. Again, the designation "a 0" indicates a value of the order of lo8 or smaller. calc a A$$) calc: a 0,00261,00262,00522,00524, ,0261,0268 A+^ ,00420, , APPLICATIONS In a preceding section, some attention has already been paid to the historically important case of the precession of Mercury's orbit, in the context of general relativity. According to Weinberg [3] there has been a great deal of interest in improving the measurements of systems

29 like this to either further support or ultimately disprove general relativity. Alternative theories to general relativity have been put forward (most notably the Brans-Dicke theory), and many of these can be summarized in the parametrized post-newtonian (PPN) formalism. The precession of an orbit under the influence of a general PPN theory is given in [3] to be where,o and 7 are two of the main parameters of the PPN formalism. For PPN to reproduce general relativity, it is proper to set /3 = 7 = 1, which in this equation is seen to return exactly the result derived previously for the general relativistic correction. Furthermore, that this expression is what yields the appropriate perihelion advance for planets like Mercury has provided a systematic check for relativity against these parametrized theories. The essence of the PPN formalism is to take the Schwarzschild metric and add terms to the coefficients of dt2 and dr2 in a systematic fashion. Beyond PPN, other theories such as that given in Rindler [8] also manipulate the basic Schwarzschild metric to calculate the effects of some new potential. In modeling an inflationary universe, it has been useful to utilize the de Sitter metric, which depends on the parameter A also arising in Einstein's field equations as the cosmological constant (originally in an attempt to guarantee a static universe). The approach of [8] was to take a "Schwarzschild-de Sitter" metric and calculate the consequences, one of which was the Newtonian energy-type equation with an additional term quadratic in r with coefficient depending on A. In the classical limit, this would correspond to an additional force proportional to r. The repulsive ("anti-gravitational") nature of this force has led to its being referred to as the effects of dark energy, which according to present models of inflation must account for 70% of the mass-energy content of the universe (with 25% accounted for by dark matter, and the remaining 5% by ordinary matter and energy). In seeing the precession of a solar system planet caused by a linear force, it was claimed by Cardona and Tejeiro[4] that this precession should be large enough, first of all, to be measurable by presently attained accuracy, and secondly to place an upper limit on the cosmelogieal-eonstant. This psssibility was ruled out directly by the work of Kerr, ffauck, and Mashhoon [7] and Wright [9], in that the actual value for the precession in each case was smaller than the uncertainty of the measurement. In the present work's agreeing formally with (71, it is agreed that no measurement of solar system data at present-day accuracy

30 TABLE W Upper Kits on A determiued by the uncertainty in precession data. Darlc energy causes a precession given by (33) for n = 1 and k = tc2. If this result were larger than the given uncertainties, this precession would have already been detected. Since no such. precession has been detected, the appropriate limit on A can be solved for by using the precession expression set equal to the uncertainty in the measurements. Mercury Venus Earth Mars Uncertainty, "/cy Amai,m"'2 1.9 x 6.7 x 2.1 x 3.88 x would be able to constrain the cosmological constant better than galactic and cosmological measurements. Using data from Pitjeva [ll] for the measurements of a precession beyond that accounted for by "known" forces (general relativity, solar oblateness, other planets, asteroids, etc.) applied to the inner planets, the cosmological constant has bounds given in Table IV. It is readily seen that, as an upper limit, these do not compete with the limit of A < 10-52m-2 quoted in Wright and Ken. Another interesting application for precession measurements is the Pioneer anomaly, given by Jaekel and Reynaud [16] and Iorio 151. The Pioneer satellites, sent to the limits of the solar system, have experienced some unknown additional acceleration of 0.8nm s"' towards the sun. This force is nearly constant between 20 and 70AU. A constant force and a linear force have already been analyzed by Iorio, who calculated the effects of such forces on the perihelia of the solar system's planets. As a demonstration of the present work's results, consider a quadratic force F(r) = kr2, where k = x reproduces the observed acceleration at T = 50AU. While Mercury and Venus still experience an undetectable precession, the force results in a measurable perihelion shift in Earth and Mars - the predicted shifts are larger than the uncertainties quoted in Table IV. The observed and calculated precessions are found in Table V. The values for Earth and Mars, however, do not agree with those actually measured. While the choice of a force dependent on r2 to explain the Pioneer anomaly is not particularly realistic, this exercise does demonstrate in principle the applicability of the present result (33) to eliminate proposed forces with precession data.

31 TABLE V: Observed and calculated values for the precession of inner planets, due to a "toy" model that attempts to explain the Pioneer anomaly. Mercury and Venus do not experience a detectable precession, but Earth and Mars do. That the calculated values do not agree with observations indicates that the proposed model should be rejected. Observed, "/cy Calculated, "Icy G The work of Dvali, Gabadadze, and Porrati [6] has put forward a possible correction to gravity arising from effects of higher-dimensional versions of the Planck mass. The effect is to add a logarithmic perturbation to the potential. This is not accommodated by the results of expression (33), since in that case n = -1 for the force law. It can, however, be calculated numerically from the expression (29). The potential is where 7-0 is the scale of the higher dimensions (called the "crossover" scale in [GI, where this logarithmic potential switches from being repulsive to attractive). Analysis of the Planck masses given in [6] yields a value for TO R! 10^m, which is about the size of the solar system. Using this parameter, however, produces unacceptably large precession values for the planets of the solar system. To obtain a more "realistic" TO, the perihelion shift for small eccentricity (22) and n = -1 is set equal to the precession uncertainty (the maximum undetectable precession) from the data of Pitjeva. The equations are solved for the parameter TO for several solar system planets. The results are in Table VI. It is interesting to see that each planet's data dictate such a great increase in the value of TO; Dvali et a1 note that this would reduce the Planck mass significantly, so that it would be possible to observe quantum gravitational effects in present-day accelerators. Because no such effects have been observed, this particular effect of higher-dimensional modifications to gravity must be ruled out. A few applications of perihelion shifts have been reviewed. The eases of general relativity and dark energy have been reviewed, and quadratic (as a "foil" explanation of the Pioneer anomaly) and logarithmic forces (arising from higher-dimensional theories) have been investigated.

32 TABLE Vk The planetary precession uncertainties are tabulated with the &mum value for TO that they would imply, based on the approximate analysis using (22). These values indicate that the logarithmic potential put forward from an analysis of higher-dimensional theories of gravity must be ruled out. Mercury Venus Earth Mars n c e t a ICY. I 0.005oI O.OOO~~ 8. CONCLUSIONS The present work has found three new expressions that approximate the shift in perihelion due to a small force in addition to gravity. The first expression (21) was found with the method used by Ohanian to derive the precession due to general relativity, and it applies only for nearly-circular orbits. The second expression (29), which is the most general, was an integral that can approximate the precession caused by any small perturbation to gravity, applied to an orbit of any eccentricity. The third expression (33) was a special case of the integral formula, applying only to forces proportional to some power of r other than rl. For small eccentricity this expression reduces to the first expression (21), when a power law force is taken (this expression is given in (22)). Because the derivation of these expressions relied on approximations that assumed terms nonlinear in the perturbations could be neglected, the expression of most practical interest (33) was checked against several tabulated results for a numerical calculation of the precession. The test proved successful, allowing the expression to be used for several applications. Several recently measured anomalies have seen attempts at explanation in terms of small additional forces, many of which take a power law form. This lends a utility to the present -results beyond a mathematical curiosity, Theexpression (33) yields results identical te those of Kerr et a1 [7] in the case of a linear force, which has been used to model dark energy. As such, this calculation lends further credence to the assertions of [7] that measurements within the solar system do not give better bounds on the cosmological constant than do