# Relativistic gravitational deflection of light and its impact on the
modeling accuracy for the Space Interferometry Mission

Slava G. Turyshev  Jet Propulsion Laboratory, California Institute of
Technology,
4800 Oak Grove Drive, Pasadena, CA 91109-0899, USA  Sternberg Astronomical
Institute, 13 Universitetskij Prospect, 119992 Moscow, Russia111Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the impact of relativistic gravitational deflection of light on the
accuracy of future Space Interferometry Mission (SIM). We estimate the
deflection angles caused by the monopole, quadrupole and octupole components
of gravitational fields for a number of celestial bodies in the solar system.
We observe that, in many cases, the magnitude of the corresponding effects is
significantly larger than the $1~{}\mu$as accuracy expected from SIM. This
fact argues for the development of a relativistic observational model for the
mission that would account for the influence of both static and time-varying
effects of gravity on light propagation. Results presented here are different
from the ones obtained elsewhere by the fact that we specifically account for
the differential nature of the future SIM astrometric measurements. We also
obtain an estimate for the accuracy of possible determination of the
Eddington’s parameter $\gamma$ via SIM global astrometric campaign; we
conclude that accuracy of $\sim 7\times 10^{-6}$ is achievable via
measurements of deflection of light by solar gravity.

Interferometric astrometry; SIM; tests of general relativity; solar system.

##  I Introduction

The last quarter of the 20th century has changed the status of Einstein’s
general theory of relativity from a purely theoretical discipline to a
practically important science. Today general relativity is the standard theory
of gravity, especially where the needs of astronomy, astrophysics, cosmology
and fundamental physics are concerned Turyshev-etal-2007 ; Turyshev-2008 ;
Soffel-etal-2003 . As such, this theory is used for many practical purposes
involving spacecraft navigation, geodesy and time transfer. Present accuracy
of astronomical observations already requires relativistic description of
light propagation as well as the relativistically correct treatment of the
dynamics of the extended celestial bodies Kopeikin-Makarov-2007 . As a result,
some of the leading static-field post-Newtonian perturbations in the dynamics
of the planets, the Moon and artificial satellites have been included in the
equations of motion, and in time and position transformation. It is also well
understood that effects due to non-stationary behavior of the solar system
gravitational field as well as its deviation from spherical symmetry should be
also considered Kopeikin-1997  and implemented in the appropriate models.

Space-based astrometry has brought about a renaissance in the entire field of
astrometry that is perhaps the most fundamental, and oldest of all areas in
astronomy Unwin-etal-2008 . The ESA Hipparcos mission, which operated from
1989-1993, yielded an astrometric catalog of 118,000 stars down to 12.5
magnitude, with positional accuracy of 1 mas for stars brighter than V = 11.
The European Space Agency (ESA) is now developing the Gaia mission as a next
generation astrometric survey mission Perryman-2001 ; Perryman-2002 , which is
expected to produce a catalog of $\sim 10^{9}$ stars, with accuracy $\simeq$
20–25 microarcsec ($\mu$as) for stars brighter than V = 15. Precision
astrometry remains a cornerstone of the field and is poised to make a major
impact on many fields of modern astronomy, astrophysics, and cosmology Unwin-
etal-2008 .

NASA’s SIM PlanetQuest mission, hereinafter SIM, as another example of a
space-based facility instrument for astrometry. The acronym SIM stands for
Space Interferometry Mission. SIM will be the first space-based Michelson
interferometer for astrometry. The instrument will operate in the optical
waveband using a 9-m baseline between the apertures. With a global astrometry
accuracy of $3~{}\mu$as for stars brighter than V = 20, it will measure
parallaxes and proper motions of stars throughout the Galaxy with
unprecedented accuracy. Operating in a narrow-angle mode, it will achieve a
positional accuracy of $0.6~{}\mu$as for a single measurement, equivalent to a
differential positional accuracy at the end of the nominal 5-year mission of =
$0.1~{}\mu$as. This performance is about 1000 times better than existing
capabilities on the ground or in space, and about 100 times better than the
upcoming Gaia mission, for differential measurements. Such high accuracy will
allow SIM to detect and measure masses of terrestrial planets around stars in
our Galactic neighborhood (see Unwin-etal-2008  for review).

SIM is a targeted mission which measures the astrometric positions of stars,
referencing the measurements to a grid of 1302 stars covering the entire sky.
Its scheduling is highly flexible, in both the order of observations, their
cadence, and the accuracy of each individual measurement. This contrasts with
the Hipparcos and Gaia missions, which scan the entire sky according to a pre-
determined scanning pattern. Many astrometry experiments can make effective
use, or in some cases require, this pointing capability for instance, searches
for terrestrial planets (especially in multiple planet systems), stellar
microlensing events, orbits of eccentric binary systems, and variable targets
such as X-ray binaries and active galactic nuclei. Currently, the ICRF,
defined by the locations of 212 extragalactic radio sources Johnston-etal-1995
; Ma-1998  with most having errors less than 1 mas, is the standard frame for
astrometry. SIM is expected to yield an optical reference frame at a level of
about $3~{}\mu$as; it will be tied’ to the ICRF by observing a number of
radio-loud quasars in common.

Figure 1: Geometry of gravitational deflection of starlight by the Sun.

In this paper, we discuss the rapidly forming field of applied general
relativity to demonstrate its influence on the high-precision astrometry.
Recent advances in the accuracy of astrometric observations have demonstrated
importance of taking into account the relativistic effects introduced by the
solar system’s gravitational environment. It is known that the reduction of
the Hipparcos data has necessitated the inclusion of stellar aberration up to
the terms of the second order in $v/c$, and the general relativistic treatment
of light bending due to the gravitational field of the Sun Perryman92  and
Earth Gould93 . Even higher modeling accuracy is anticipated for Gaia
Klioner-2004 .

Prediction of the gravitational deflection of light was one of the first
successes of general relativity. Since the first confirmation by the
Eddington’s expedition in 1919, the effect of gravitational deflection has
been studied quite extensively and currently analysis of almost every precise
astronomical measurement must take this effect into account Modest96 ;
Brumberg-1972 ; Brumberg-1991 ; Turyshev-etal-2007 ; Turyshev-2008 . According
to general relativity, the light rays propagating near a gravitating body are
achromatically deflected by the body’s relativistic gravity field. The whole
trajectory of the light ray is bent towards the body by an angle depending on
the strength of the body’s gravity. The solar gravity field produces the
largest effect on the light traversing the solar system. To first order in the
gravitational constant, $G$, the solar deflection angle $\theta^{\odot}_{\tt
gr}$ depends only on the solar mass $M_{\odot}$ and the impact parameter $d$
relative to the Sun Will_book93 :

$\theta^{\odot}_{\tt gr}={4GM_{\odot}\over
c^{2}d}\cdot\frac{1+\cos\chi}{2}=1^{\prime\prime}.751\Big{[}\frac{R_{\odot}}{d}\Big{]}\frac{1+\cos\chi}{2},$
(1)

where $R_{\odot}$ solar radus. The absolute magnitude for the light deflection
angle is maximal for the rays grazing the sun, e.g. $\theta^{\odot}_{\tt
gr}=1.751$ seconds of arc. Most of the measurements of the gravitational
deflection to date involved the solar gravity field, planets in the solar
system or gravitational lenses. For the future astrometric observations with
SIM, in addition to the Sun, effect of planetary gravitatiobal deflections of
light must also be considered. The most precise measurement of the light
deflection with the planet Jupiter was done in Fomalont-Kopeikin03 .
Relativistic deflection of light has been observed, with various degrees of
precision, on distance scales of $10^{9}$ to $10^{21}$ m, and on mass scales
from $10^{-3}$ to $10^{13}$ solar masses, the upper ranges determined from the
gravitational lensing of quasars Dar92 ; TreuhaftLowe91 .

In the case of SIM, the star is assumed to be at a very large distance
compared to the Sun, and $\chi$ is the angular separation between the
deflector and the star. With the space observations carried out by SIM, $\chi$
is not necessarily a small angle. The relevant geometry and notations are
shown in Fig. 1. In this figure we emphasized the fact that the difference of
the apparent position of the source from it’s true position depends on the
impact parameter of the incoming light with respect to the deflector. For the
astrometric accuracy of a few $\mu$as and, in the case when the Sun is the
deflector, positions of all observed sources experiencing such a displacement.
This is why, in order to correctly account for the effect of gravitational
deflection, it is important to process together the data taken with the
different separation angles from the deflector. In the wide-angle astrometry
mode SIM, will be observing the sky in a 15∘ patches of sky (called field of
regard or FoR) making a set of differential observations within the FoR.
Therefore, this differential nature of the measurements would result in
minimizing the contribution of the gravitational deflection on the single
measurement. To reflect this fact, we will present results for two types of
astrometric measurements, namely for the absolute (single ray deflection) and
differential (two sources separated by the 15∘ field of regard) observations.

A major objective of this paper is to show that, before microarcsecond-level
astrometry will become a powerful tool for 21-st century astronomy, there is a
need for an adequate modeling necessary to match this new frontier of
astrometric accuracy. The prospect of new high precision astrometric
measurements from space with SIM requires inclusion of relativistic effects at
the $(v/c)^{3}$ level Turyshev98 ; Unwin-etal-2008 . At the level of accuracy
expected from SIM, even more subtle gravitational effects on astrometry from
within the solar system will start to become apparent, such as the monopole
and the quadrupole components of the gravitational fields of the planets
Sovers98  and the gravito-magnetic effects caused by their motions and
rotations. Thus, the identification of all possible sources of “astrophysical”
noise that may contribute to the future SIM astrometric campaign, is well
justified.

This work is organized as follows: Section II discusses the influence of the
relativistic deflection of light by the monopole components of the
gravitational fields of the solar system’s bodies. We present the model and
our estimates for the most important effects that will be influencing
astrometric observations of a few $\mu$as accuracy, that will be made from
within the solar system. Section III will specifically address three most
intense gravitational environments in the solar system, namely the vicinities
of the Sun, Jupiter and Earth. In Section IV we will discuss the effects of
the gravitational deflection of light by the higher gravitational multipoles
(both mass and current ones) of some of the bodies in the solar system. We
derive constraints on the navigation of the spacecraft and the accuracy of the
solar system ephemerides. In Section V we investigate the possibility of
improving the accuracy of the Eddington’s parameter $\gamma$ via astrometric
tests of general relativity in the solar system. We also discuss the
opportunity to measure the solar acceleration towards the Galactic Center with
SIM. We will conclude the paper with the discussion of the results obtained
and our recommendations for future studies.

##  II Gravity Contributions to the Local Astrometric Environment

In this Section we develop a model for light propagation that will be used to
estimate various relativistic effects due to gravitational deflection of light
by the solar system’s bodies.

###  II.1 Relativistic deflection of light by the gravity monopole

The first step into a relativistic modeling of a light path consists of
determining the direction of the incoming photon as measured by an observer
located in the solar system as a function of the barycentric coordinate
position of the light source. Apart from second and third orders of velocity
aberration the only other sizable effect is due to the bending of light rays
in the gravitational field of solar system bodies Brumberg-Klioner-
Kopeikin-1990 ; Turyshev98 . Effects of the gravitational monopole deflection
of light are the largest among those in the solar system.

Generalizing on a phenomenological parameterization of the gravitational
metric tensor field, which Eddington originally developed for a special case,
a method called the parameterized post-Newtonian (PPN) formalism has been
developed (see Turyshev-2008  for discussion). This method represents the
gravity tensor’s potentials for slowly moving bodies and weak inter-body
gravity, and is valid for a broad class of metric theories, including general
relativity as a unique case. The several parameters in the PPN metric
expansion vary from theory to theory, and they are individually associated
with various symmetries and invariance properties of the underlying theory
(see Will_book93  for details).

If (for the sake of simplicity) one assumes that Lorentz invariance, local
position invariance and total momentum conservation hold, the metric tensor
for a single, slowly-rotating gravitational source is given by:

$\displaystyle g_{00}$ $\displaystyle=$ $\displaystyle
1-2\frac{GM}{c^{2}r}\Big{(}1-J_{2}\frac{R^{2}}{r^{2}}\frac{3\cos^{2}\theta-1}{2}\Big{)}+{\cal
O}(c^{-4}),$ $\displaystyle g_{0i}$ $\displaystyle=$ $\displaystyle
2(\gamma+1)\frac{G[\vec{\cal S}\times\vec{r}]_{i}}{c^{3}r^{3}}+{\cal
O}(c^{-5}),$ (2) $\displaystyle g_{ij}$ $\displaystyle=$
$\displaystyle-\delta_{ij}\Big{[}1+2\gamma\frac{GM}{c^{2}r}\Big{(}1-J_{2}\frac{R^{2}}{r^{2}}\frac{3\cos^{2}\theta-1}{2}\Big{)}+{\cal
O}(c^{-5}),$

where $M$ and $\vec{\cal S}$ being the mass and angular momentum of the body,
$J_{2}$ and $R$ are the body’s quadrupole moment and its radius, and $G$ is
the universal gravitational constant, $r$ is the distance from the center of
the body to a particular point. The $1/c^{2}$ term in $g_{00}$ is the
Newtonian limit; the $1/c^{3}$ term in $g_{0i}$ and the $1/c^{2}$ term in
$g_{ij}$, are post-Newtonian corrections. All of these terms are required to
describe light propagation phenomena to the first post-Newtonian order.

The Eddington parameter $\gamma$ in the Eqs. (II.1) represents the measure of
the curvature of the space created by a unit rest mass Will_book93 . Note that
general relativity, when analyzed in standard gauge of the PPN formalism (see
Will_book93 ; Will-lrr-2006-3  for details), gives: $\gamma=1$. The Brans-
Dicke theory is the most famous among the alternative theories of gravity. It
contains, besides the metric tensor, a scalar field $\phi$ and an arbitrary
coupling constant $\omega$, related to this PPN parameter as
$\gamma={1+\omega\over 2+\omega}$. The stringent observational bound resulting
from the 2003 experiment with the Cassini spacecraft require that
$|\omega|\gtrsim 40000$ Bertotti-Iess-Tortora-2003 ; Will-lrr-2006-3 . There
exist additional alternative theories that provide guidance for gravitational
experiments Will-lrr-2006-3 .

Metric tensor Eq. (II.1) can be used to derive expressions need to describe
propagation of electro-magnetic signals between any of the two points in
space. Following the standard procedure of integrating light geodesics (see
Brumberg-1972 ; Brumberg-1991  for details), the corresponding light-time
equation for a single deflecting body can be derived in the following form

$t_{2}-t_{1}=\frac{r_{12}}{c}+(1+\gamma)\frac{GM}{c^{3}}\ln\left[\frac{r_{1}+r_{2}+r_{12}}{r_{1}+r_{2}-r_{12}}\right]+{\cal
O}(c^{-5}),$ (3)

where $t_{1}$ refers to the signal transmission time, and $t_{2}$ refers to
the reception time. $r_{1,2}$ are the barycentric positions of the transmitter
and receiver, and $r_{12}$ is their spatial separation (see Moyer-2003  for
details). Also, for the moment, we neglected the presence of $J_{2}$ and
$\vec{\cal S}$ in the Eqs. (II.1), but will investigate contributions of these
and other multipoles in Sec. IV.

For a realistic observing scenario with the SIM, the sources of light are
located far out side the solar system, $r_{2}\ll r_{1}\equiv r_{S}$ and
$r_{12}$ can be approximated as $r_{12}=|\vec{r}_{1}-\vec{r}_{2}|\simeq
r_{1}-(\vec{n}_{S}\cdot\vec{r}_{2})$, where we introduced a notation
$\vec{n}_{S}=\vec{r}_{1}/r_{1}$. This approximation allows one to represent
the expression in the square brackets in Eq. (3) as follows:

$\frac{r_{1}+r_{2}+r_{12}}{r_{1}+r_{2}-r_{12}}\simeq\frac{2r_{1}+r_{2}-(\vec{n}_{S}\cdot\vec{r}_{2})}{r_{2}+(\vec{n}_{S}\cdot\vec{r}_{2})}.$
(4)

Figure 2: Basic geometry of light propagation in the stellar interferometry.

The SIM instrument does not directly measure the angular separation between
stars, but the projection of each star direction vector onto the
interferometer baseline by measuring the pathlength delay of starlight as it
passes through the two arms of the interferometer. The SIM instrument will
precisely measure optical path difference (OPD) between the wavefronts of
light received by the two telescopes forming the interferometric baseline (see
Fig. 2). (In fact, for SIM, with its 10 m baseline, such a measurement will be
done with a precision at the level of 1 picometer.) This difference will
result in the different internal OPDs needed to apply in order to coherently
add the signals. The delay measurement is made by a combination of internal
metrology measurements to determine the distance the starlight travels through
each arm, external metrology measurements that determine the length and local
orientation of the baseline, and a measurement of the central white light
fringe to determine the point of equal optical pathlength Unwin-etal-2008 .
Therefore, the OPD is the main observable that the interferometric instrument
will measure; relativistic modeling this delay will be among the main
objectives of the upcoming SIM modeling effort.

An interferometer measures optical path difference (OPD) between the
wavefronts of light received by the two telescopes forming the interferometric
baseline $\vec{b}$ at points $\vec{r}_{2}$ and
$\vec{r}^{\prime}_{2}=\vec{r}_{2}+\vec{b}$. To account for that fact we need
to determine the temporal difference between the signals received at these
telescopes which is
$\ell=c(t^{\prime}_{2}-t_{1})-c(t_{2}-t_{1})=c(t^{\prime}_{2}-t_{2})$. The
first term in Eq. (3) is the geometric delay. Using approximations $(b\ll
r_{2}\ll r_{1})$, it is easy to see that this term leads to the approximate
expression for geometric delay, given as $\ell_{\rm
geom}=r_{12}^{\prime}-r_{12}=-(\vec{b}\cdot\vec{n}_{S})/(1-\vec{n}_{S}\cdot\dot{\vec{r}}_{2}/c)$.
In this paper, we concern with only the largest contributions from the
gravitational defection of light, thus most of the velocity-dependent terms
will discarded (see Sovers98 ; Kopeikin-Schafer-1999  for details).

The second term Eq. (3) is the relativistic delay $\ell_{\rm gr}$, the focus
of this work. Using Eq. (4) one can present expression in the square brackets
of Eq. (3) as below

$\frac{2r_{1}+{r_{2}}^{\prime}-(\vec{n}_{S}\cdot\vec{r}_{2}{}^{\prime})}{2r_{1}+r_{2}-(\vec{n}_{S}\cdot\vec{r}_{2})}~{}\frac{r_{2}+(\vec{n}_{S}\cdot\vec{r}_{2})}{{r_{2}}{}^{\prime}+(\vec{n}_{S}\cdot\vec{r}_{2}{}^{\prime})}\simeq
1-\frac{1}{r_{2}}\,\frac{\vec{b}(\vec{n}_{S}+\vec{n}_{2})}{1+(\vec{n}_{S}\cdot\vec{n}_{2})}.$
(5)

In the first order in gravitational constant, one can add individual
interferometric delays due to the gravity of the bodies along the light path.
As a result, the general relativistic contribution to the OPD $\ell_{\tt
gr}=c\tau_{\tt gr}$ takes the following approximate form:

$\ell_{\tt
gr}=-(\gamma+1)\sum_{B}\frac{G}{c^{2}}\frac{M_{B}}{r_{B}}\Big{[}{\vec{b}(\vec{n}_{S}+\vec{n}_{B})\over
1+(\vec{n}_{S}\cdot\vec{n}_{B})}\Big{]},$ (6)

where $r_{B}$ is the distance from SIM to a deflecting body $B$,
$\vec{n}_{B}={\vec{r}}_{B}/r_{B}$ is the unit vector in this direction. This
OPD is the leading general relativistic observable that the interferometric
instrument will measure; a complete relativistic modeling this delay should be
the main objective of the upcoming SIM modeling effort.

In general, a three-dimensional approach must be used in order to work out a
practical model of the interferometric time delay. Howevere, for the purpose
of this paper, it is sufficient to confine our analysis to a plane and
parameterize the quantities involved as follows (see Fig. 3):

$\vec{b}=b\,(\cos\epsilon,\,\sin\epsilon),~{}~{}~{}~{}~{}\vec{r}_{B}=r_{B}\,(\cos\alpha_{B},\,\sin\alpha_{B}),~{}~{}~{}~{}~{}\vec{n}_{S}=(\cos\theta,\,\sin\theta),$
(7)

where $\epsilon$ is the angle of the baseline’s orientation with respect to
the instantaneous body-centric coordinate frame, $\alpha_{B}$ is the right
assention angle of the interferometer as seen from the this frame and $\theta$
is the direction to the observed source correspondingly. The geometry of the
problem and notations are presented in the Figure 3.

Figure 3: Geometry and notations for the gravitational deflection of light.

It is convenient to express the gravitational contribution to the total OPD
Eq.(6) in terms of the deflector and the source separation angle $\chi_{B}$ as
observed by the interferometer. In our approximation, the following relations
$d=r_{S}\sin\chi_{S}=r_{B}\sin\chi_{B}$ and
$\chi_{S}+\chi_{B}+\theta-\alpha=\pi$ are valid; this allows one to eliminate
angle $\chi_{S}$ by expressing the source’s position angle $\theta$ via the
separation angle $\chi_{B}$ as below:

$\theta=\pi+\alpha_{B}-\chi_{B}-\arcsin\Big{[}\,\frac{r_{B}}{r_{S}}\,\sin\chi_{B}\,\Big{]}.$
(8)

As the sources will be located at a very large distance, $r_{S}$, compares to
the distance between the interferometer and the deflector ($r_{B}\ll r_{S}$),
we can neglect the presence of the last term in the equation Eq.(8), so that
$\theta\simeq\pi+\alpha_{B}-\chi_{B}$. After substituting expressions (8) and
(7) into Eq. (6), we rewrite the contribution of the gravitational deflection
to the total OPD, Eq.(6), in the following form:

$\ell_{\tt
gr}=-(\gamma+1)\sum_{B}\frac{G}{c^{2}}\frac{M_{B}b}{r_{B}}\Big{[}\cos(\epsilon-\alpha_{B})+\sin(\epsilon-\alpha_{B})\frac{1+\cos\chi_{B}}{\sin\chi_{B}}\Big{]}.$
(9)

This expression describes gravitational delay as measured by an
interferometer; we will use it for estimation purposes.

####  II.1.1 Absolute Astrometric Measurements

Eq. (9) is appropriate for estimation the magnitudes of the gravitational
bending effects measured by an interferometer. It depends on the angle between
the baseline and deflector-instrument vectors, $\epsilon-\alpha_{B}$. As the
main objective of this paper is to estimate the magnitudes of the effects
involved, we choose $\epsilon-\alpha_{B}=\frac{\pi}{2}$ that maximizes
contribution of each individual deflector for a particular orbital position of
the spacecraft and the baseline orientation. As a result, in the SIM proper
reference frame Eq. (9) may be re-written as $\ell_{\tt
gr}=-\sum_{B}\,\ell_{\tt gr}^{B}$, with the individual contributions of the
deflecting bodies to gravitational delay $\ell_{\tt gr}^{B}$ and deflection
angle $\theta^{B}_{\tt gr}\simeq{\ell^{B}_{\tt gr}}/{b}$ in the following form

$\ell_{\tt
gr}^{B}=-(\gamma+1)\frac{G}{c^{2}}\frac{M_{B}b}{r_{B}}\frac{1+\cos\chi_{B}}{\sin\chi_{B}}\qquad{\rm
and}\qquad\theta^{B}_{\tt
gr}=-(\gamma+1)\frac{G}{c^{2}}\frac{M_{B}}{r_{B}}\frac{1+\cos\chi_{B}}{\sin\chi_{B}}.$
(10)

The two expressions $\ell_{\tt gr}^{B}$ and $\theta^{B}_{\tt gr}$ will be used
interchangeably throughout the paper.

For complete analysis of the gravitational deflection of light we will have to
account for the time dependency in all the quantities involved. Thus, one will
have to use the knowledge of the position of the spacecraft in the solar
system’s barycentric reference frame, the instrument’s orientation in the
proper coordinate frame Klioner03 , the time that was spent in a particular
orientation, the history of all the maneuvers and re-pointings of the
instrument, etc. These issues are closely related to the principles of the
operational mode of the instrument that is currently still being developed.

####  II.1.2 Differential Astrometric Measurements

SIM will perform its astrometric campaign working in differential mode either
within FoR=15∘ for the wide angle astrometry or within FoR=1∘ for the narrow
angle astrometry. To evaluate the impact of gravitational delay of light on
these measurements, we need to derive the appropriate expressions reflecting
the differential nature of astrometric measurements with SIM.

Within the accepted approximation, the necessary expression for the
differential OPD may be obtained by subtracting OPDs for the different sources
one from one another. Using Eq. (9), this results in the following expression:

$\delta\ell^{B}_{\tt gr}=\ell^{B}_{1\tt gr}-\ell^{B}_{2\tt
gr}=-(\gamma+1)\sum_{B}\frac{G}{c^{2}}\frac{M_{B}}{r_{B}}\Big{[}\frac{\vec{b}(\vec{n}_{S1}+\vec{n}_{B})}{1+(\vec{n}_{S1}\vec{n}_{B})}-\frac{\vec{b}(\vec{n}_{S2}+\vec{n}_{B})}{1+(\vec{n}_{S2}\vec{n}_{B})}\Big{]},$
(11)

where ${\vec{n}}_{S1}$ and $\vec{n}_{S2}$ are the barycentric positions of the
primary and the secondary objects. By using parameterization for the
quantities involved similar to that above ($b\ll r_{B}\ll r_{S1,S2}$), this
expression may be presented in terms of the deflector-source separation
angles, $\chi_{1B},~{}\chi_{2B}$, as follows:

$\delta\ell_{\tt
gr}=-(\gamma+1)\sum_{B}\frac{G}{c^{2}}\frac{M_{B}b}{r_{B}}\sin(\epsilon-\alpha_{B})\,\,\frac{\sin\frac{1}{2}(\chi_{B2}-\chi_{B1})}{\sin\frac{1}{2}\chi_{B1}\sin\frac{1}{2}\chi_{B2}}.$
(12)

Similar to the discussion of absolute defection angles, we choose
$\epsilon-\alpha_{B}=\frac{\pi}{2}$ that maximizes contribution of each
individual deflector for a particular orbital position of the spacecraft and
the baseline orientation. Therefore, in the SIM proper reference frame Eq.
(12) may be re-written as $\delta\ell_{\tt gr}=-\sum_{B}\,\delta\ell_{\tt
gr}^{B}$, with the individual contributions for gravitational delay
$\delta\ell_{\tt gr}^{B}$ and corresponding deflection angle
$\delta\theta^{B}_{\tt gr}\simeq\delta{\ell^{B}_{\tt gr}}/{b}$ in the
following form

$\delta\ell^{B}_{\tt
gr}=(\gamma+1)~{}\frac{G}{c^{2}}\frac{M_{B}b}{r_{B}}~{}\frac{\sin\frac{1}{2}(\chi_{2B}-\chi_{1B})}{\sin\frac{1}{2}\chi_{1B}\cdot\sin\frac{1}{2}\chi_{2B}}\qquad{\rm
and}\qquad\delta\theta^{B}_{\tt
gr}=(\gamma+1)~{}\frac{G}{c^{2}}\frac{M_{B}}{r_{B}}~{}\frac{\sin\frac{1}{2}(\chi_{2B}-\chi_{1B})}{\sin\frac{1}{2}\chi_{1B}\cdot\sin\frac{1}{2}\chi_{2B}}.$
(13)

The two expressions $\delta\ell^{B}_{\tt gr}$ and $\delta\theta^{B}_{\tt gr}$
will be used interchangeably throughout the paper.

###  II.2 Deflection of Grazing Rays by the Bodies of the Solar System

Table 1: Relativistic monopole deflection of by the solar system bodies at the SIM’s location.  Solar | Angular size | Deflection of grazing rays
---|---|---
system’s | at SIM pos., | absolute | diff. $[15^{\circ}]$ | diff. $[1^{\circ}]$
object |  ${\cal R}_{B}$, arcsec |  $\theta^{B}_{\tt gr},\mu$as |  $\delta\theta^{B}_{\tt gr},~{}\mu$as |  $\delta\theta^{B}_{\tt gr},~{}\mu$as
Sun | 0∘.26656 | 1′′.75064 | 1′′.72025 | 1′′.38221
Sun at 45∘ | 45∘ | 9 831.39 | 2 777.97 | 237.66
Moon | 47.92690 | 25.91 | 25.87 | 25.56
Mercury | 5.48682 | 82.93 | 82.92 | 82.81
Venus | 30.15040 | 492.97 | 492.69 | 488.88
Earth | 175.88401 | 573.75 | 571.90 | 547.03
Mars | 8.93571 | 115.85 | 115.83 | 115.57
Jupiter | 23.23850 | 16 419.61 | 16 412.60 | 16 314.30
Jupiter at 30′′ | 30.0 | 12 719.12 | 12 712.03 | 12 614.21
Saturn | 9.64159 | 5 805.31 | 5 804.27 | 5 789.79
Uranus | 1.86211 | 2 171.38 | 2 171.30 | 2 170.26
Neptune | 1.18527 | 2 500.35 | 2 500.29 | 2 499.52
Pluto | 0.11478 | 2.82 | 2.82 | 2.82

We are now ready to evaluate the influence of the solar system’s gravity field
on the future high-accuracy astrometric observations. In particular, we
estimate the magnitudes of the angles of gravitational deflection for those
light rays that are grazing the surfaces of celestial bodies.

Table 1 shows magnitude of the angles characterizing relativistic monopole
deflection of grazing (e.g. $\chi_{1B}={\cal R}_{B}$) light rays by the bodies
of the solar system at the SIM’s location (i.e., the solar Earth-trailing
orbit Unwin-etal-2008 ). Results for absolute deflection angles agree with
values obtained by other authors (for instance, Brumberg-Klioner-Kopeikin-1990
). The results presented in the terms of the following quantities:

  * i).

for absolute astrometry results are given in terms of the absolute
measurements $\ell^{B}_{\tt gr}$ and $\theta^{B}_{\tt gr}$ from Eq. (10);

  * ii).

for differential astrometry results are given in terms of the absolute
measurements $~{}\delta\ell^{B}_{\tt gr}$ and $\delta\theta^{B}_{\tt gr}$ from
Eq. (13).

For the differential observations the two stars are assumed to be separated by
the size of the instrument’s field of regard. For the grazing rays, position
of the primary star is assumed to be on the limb of the deflector. Moreover,
results are given for the smallest distances from SIM to the bodies (e.g.when
the gravitational deflection effect is largest). For the Earth-Moon system we
took the SIM’s position at the end of the first half of the first year mission
at the distance of 0.05 AU from the Earth. Presented in the right column of
Table 1 are the magnitudes of the body’s individual contributions to the
gravitational delay of light at the SIM’s location.

Note, that the angular separation of the secondary star will always be taken
larger than that for the primary. It is convenient to study the case of the
most distant available separations of the sources. In the case of SIM, this is
the size of the field of regard (FoR). Thus for the wide-angle astrometry the
size of FoR will be $15^{\circ}\equiv\frac{\pi}{12}$ rad, thus
$\chi_{2B}=\chi_{1B}+\frac{\pi}{12}$. For the narrow-angle observations this
size is FoR $=1^{\circ}\equiv\frac{\pi}{180}$ rad, thus for this type of
astrometric observations we will use $\chi_{2B}=\chi_{1B}+\frac{\pi}{180}$;
finally, $b=10$ m is the baseline length used in the estimates.

####  II.2.1 Critical Impact Parameter for High Accuracy Astrometry

The estimates, presented in the Table 1 have demonstrated that it is very
important to correctly model and account for gravitational influence of the
bodies of the solar system. Depending on the impact parameter $d_{B}$ (or
planet-source separation angle, $\chi_{B}$), one will have to account for the
post-Newtonian deflection of light by a particular planet. Most important is
that one will have to permanently monitor the presence of some of the bodies
of the solar system during all astrometric observations, independently on the
position of the spacecraft in it’s solar orbit and the observing direction.
The bodies that introduce the biggest astrometric inhomogeneity are the Sun,
Jupiter and the Earth (especially at the beginning of the mission, when the
spacecraft is in the Earth’ immediate proximity).

Table 2: Relativistic monopole deflection of light: the angles and the critical distances for $\Delta\theta_{0}=1~{}\mu$as astrometric accuracy. Solar defection of light must be always taken into account, as at the SIM’s position at 1 AU from the Sun, the solar gravitational deflection effects are always larger than 1 $\mu$as. The critical distances for the Earth are given for two distances, namely for 0.05 AU (27∘.49) and 0.01 AU (78∘.54).  Object |  $\theta^{B}_{\tt gr},~{}\mu$as | Critical distances for accuracy of $1~{}\mu$as
---|---|---
|  |  $d^{B}_{\tt crit}$, km |  $d^{B}_{\tt crit}$, deg |  $d^{B}_{\tt crit}$, ${\cal R}_{B}$
Sun | 1′′.75064 | always | always | always
Moon | 25.91 | $4.501\times 10^{4}$ | $0^{\circ}.34-1^{\circ}.72$ | $25.9\cdot{\cal R}_{m}$
Mercury | 82.93 | $2.023\times 10^{4}$ | $0^{\circ}.06-0^{\circ}.13$ | $82.9\cdot{\cal R}_{Me}$
Venus | 492.97 | $2.982\times 10^{6}$ | $0^{\circ}.66-4^{\circ}.13$ | $492.9\cdot{\cal R}_{V}$
Earth | 573.75 | $3.453\times 10^{6}$ | 27${}^{\circ}.49-78^{\circ}$.54 | $541.4\cdot{\cal R}_{\oplus}$
Mars | 115.85 | $3.931\times 10^{5}$ | 0${}^{\circ}.06-0^{\circ}$.29 | $115.9\cdot{\cal R}_{Ms}$
Jupiter | 16 419.61 | $6.270\times 10^{8}$ | $64^{\circ}.06-88^{\circ}.51$ | $8\,849\cdot{\cal R}_{J}$
Saturn | 5 805.31 | $3.420\times 10^{8}$ | $12^{\circ}.56-15^{\circ}.45$ | $5\,700\cdot{\cal R}_{S}$
Uranus | 2 171.38 | $5.319\times 10^{7}$ | $1^{\circ}.01-1^{\circ}.12$ | $2\,171\cdot{\cal R}_{U}$
Neptune | 2 500.35 | $6.276\times 10^{7}$ | $0^{\circ}.77-0^{\circ}.82$ | $2\,500\cdot{\cal R}_{N}$
Pluto | 2.82 | $9.025\times 10^{3}$ | $0^{\prime\prime}.31-0^{\prime\prime}.32$ | $2.8\cdot{\cal R}_{P}$

Let us introduce a measure of such a gravitational inhomogeneity due to a
particular body in the solar system. To do this, suppose that future
astrometric experiments with SIM will be capable to measure astrometric
parameters with accuracy of $\Delta\theta_{0}=\Delta k~{}\mu$as, where $\Delta
k$ is some number characterizing the accuracy of the instrument (e.g. for a
single measurement accuracy $\Delta k=3$ for stars brighter than $V=20$ and
for the mission accuracy $\Delta k=0.1$, see Unwin-etal-2008 ). Then, there
will be a critical distance from the body, beginning from which, it is
important to account for the presence of the body’s gravity in the vicinity of
the observed part of the sky. We call this distance – critical impact
parameter, $d^{B}_{\tt crit}$, the closest distance between the body and the
light ray that is gravitationally deflected to the angle

$\theta^{\tt c}_{\tt gr}(d^{B}_{\tt crit})=\Delta\theta_{0}=\Delta k~{}\mu{\sf
as}.$ (14)

The necessary expression for $d^{B}_{\tt crit}$ is obtained from Eq. (10).
Assuming that the angle $\chi_{B}$ is small and noting that
$r_{B}\sin\chi_{B}=d$, we can write this equation as follows $\theta^{B}_{\tt
gr}\simeq{2\mu_{B}}/{d}$, where $\mu_{B}=2GM_{B}/c^{2}$ being relativistic
gravitational radius of the body. As the effect of gravitational deflection
light is inversely proportional to the impact parameter, then beginning from a
certain value of the parameter, $d^{B}_{\tt crit}$, the deflection angle will
be larger $\Delta\theta_{0}$; this value is given by the following expression:

$d^{B}_{\tt crit}=\frac{2\,\mu_{B}}{\Delta\theta_{0}},$ (15)

Different forms of the critical impact parameters $d^{B}_{\tt crit}$ for
$\Delta\theta_{0}=1~{}\mu$as are given in the Table 2. With the help of Eq.
(15), the results given in this table are easily scaled for any astrometric
accuracy $\Delta\theta_{0}$.

####  II.2.2 Deflection of Light by Planetary Satellites

One may expect that the planetary satellites will affect the astrometric
studies if a light ray would pass in their vicinities. Just for completeness
of our study we would like to present the estimates for the gravitational
deflection of light by the planetary satellites and the small bodies in the
solar system. The corresponding estimates for deflection angles,
$\theta^{B}_{\tt gr}$, and critical distances, $d_{\tt crit}$ are presented in
the Table 3. Due, to the fact that the angular sizes for those bodies are much
less than the smallest field of regard of the SIM instrument (e.g. FoR=1∘),
the results for the differential observations will be effectively insensitive
to the size of the the two available FoRs. The obtained results demonstrate
the fact that observations of these objects with that size of FoR will
evidently have the effect from the relativistic bending of light. Thus, in
Table 3 we have presented there only the angle for the absolute gravitational
deflection in terms of quantities $\theta^{B}_{\tt gr}$. As a result, the
major satellites of Jupiter, Saturn and Neptune should also be included in the
model if the light ray passes close to these bodies.

Table 3: Relativistic deflection of light by some planetary satellites. Object | Mass, | Radius, | Angular size, | Grazing | 1 $\mu$as critical radius
---|---|---|---|---|---
|  $10^{25}$ g |  ${\cal R}_{B}$, km |  ${\cal R}_{B}$, arcsec |  $\theta^{B}_{\tt gr},~{}\mu$as |  $d_{\tt crit},$ km |  $d_{\tt crit},$ ${\cal R}_{planet}$
Io | 7.23 | 1 738 | 0.570056 | 25.48 | 44 291 | $0.63\cdot{\cal R}_{J}$
Europa | 4.7 | 1 620 | 0.531353 | 17.77 | 28 793 | $0.41\cdot{\cal R}_{J}$
Ganymede | 15.5 | 1 415 | 0.464114 | 67.11 | 94 954 | $1.34\cdot{\cal R}_{J}$
Callisto | 9.66 | 2,450 | 0.803589 | 24.15 | 59 178 | $0.84\cdot{\cal R}_{J}$
Rhea | 0.227 | 675 | 0.108468 | 2.06 | 1 391 | $0.02\cdot{\cal R}_{S}$
Titan | 14.1 | 2 475 | 0.397715 | 34.90 | 86 378 | $1.44\cdot{\cal R}_{S}$
Triton | 13 | 1 750 | 0.082638 | 45.51 | 79 639 | $3.17\cdot{\cal R}_{N}$

####  II.2.3 Gravitational Influence of Small Bodies

Additionally, for the astrometric accuracy at the level of few $\mu$as (i.g.,
$\Delta\theta_{0}=\Delta k~{}\mu{\rm as}$), one needs to account for the post-
Newtonian deflection of light due to rather a large number of small bodies in
the solar system having a mean radius

${\cal R}_{B}\geq 624~{}\sqrt{\Delta k\over\rho_{B}}\hskip 8.0pt~{}{\rm km}.$
(16)

The deflection angle for the largest asteroids Ceres, Pallas and Vesta for
$\Delta k=1$ are given in the Table 4. The quoted properties of the asteroids
were taken from StandishHellings89 ; Mouret-Hestroffer-Mignard-2007 .
Positions of these asteroids are known and they are incorporated in the JPL
ephemerides. Other small bodies (e.g. asteroids, Kuiper belt objects, etc.)
may produce a stochastic noise in the future astrometric observations with
SIM; therefore, they should also be properly modeled.

Table 4: Relativistic deflection of light by the asteroids.  Object |  $\rho_{B},$ g/cm3 | Radius, km |  $\theta^{B}_{\tt gr},~{}\mu$as
---|---|---|---
Ceres | 2.3 | 470 | 1.3
Pallas | 3.4 | 269 | 0.6
Vesta | 3.6 | 263 | 0.6
Class S | 2.1 $\pm$ 0.2 | TBD | $\leq 0.3$
Class C | 1.7 $\pm$ 0.5 | TBD | $\leq 0.3$

##  III Regions with the most gravitationally intense environments for SIM

The properties of the solar system’s gravity field presented in the Tables 1
and 2 suggest that the most intense gravitational environments in the solar
system are those offered by the Sun and two planets, namely the Earth and
Jupiter. In this Section we will analyze these regions in more details.

###  III.1 Gravitational Deflection of Light by the Sun

From the expressions Eq. (9) and Eq. (13) we obtain the relations for
relativistic deflection of light by the solar gravitational monopole. The
expression for the absolute astrometry takes the form:

$\theta^{\odot}_{\tt gr}=(\gamma+1)~{}\frac{G}{c^{2}}\frac{M_{\odot}}{r_{\rm
AU}}~{}\frac{1+\cos\chi_{1\odot}}{\sin\chi_{1\odot}}=4.072\cdot\frac{1+\cos\chi_{1\odot}}{\sin\chi_{1\odot}}~{}~{}~{}~{}{\rm
mas},$ (17)

where $\chi_{1\odot}$ is the Sun-source separation angle, $r_{\rm AU}=1$ AU,
and $\gamma=1$. Similarly, for differential astrometric observations one
obtains:

$\delta\theta^{\odot}_{\tt
gr}=(\gamma+1)\frac{G}{c^{2}}~{}\frac{M_{\odot}}{r_{\rm
AU}}\frac{\sin\frac{1}{2}(\chi_{2\odot}-\chi_{1\odot})}{\sin\frac{1}{2}\chi_{1\odot}\cdot\sin\frac{1}{2}\chi_{2\odot}}=4.072\cdot\frac{\sin\frac{1}{2}(\chi_{2\odot}-\chi_{1\odot})}{\sin\frac{1}{2}\chi_{1\odot}\cdot\sin\frac{1}{2}\chi_{2\odot}}~{}~{}~{}~{}{\rm
mas},$ (18)

with $\chi_{1\odot},\chi_{2\odot}$ being the Sun-source separation angles for
the primary and the secondary stars correspondingly. We use two stars
separated by the SIM’s field of regard, namely
$\chi_{2\odot}=\chi_{1\odot}+\frac{\pi}{12}$. The solar angular dimensions
from the Earth’ orbit are calculated to be ${\cal R}_{\odot}=0^{\circ}.26656$.
This angle corresponds to a deflection of light to $1.75065$ arcsec on the
limb of the Sun. Results for the most interesting range of $\chi_{1\odot}$ are
given in the Table 5.

Table 5: Magnitudes of the gravitational deflection angle vs. the Sun-source
separation angle $\chi_{1\odot}$.

Solar | small $\chi_{1\odot},$ deg
---|---
deflection | $0^{\circ}.27$ | $0^{\circ}.5$ | $1^{\circ}$ | $2^{\circ}$ | $5^{\circ}$ | $10^{\circ}$ | $15^{\circ}$
$~{}\theta^{\odot}_{\tt gr}$, mas | 1 728 | 933.295 | 466.639 | 233.302 | 93.271 | 46.547 | 30.932
$\delta\theta^{\odot}_{\tt gr}~{}[15^{\circ}]$, mas | 1 698 | 903.372 | 437.663 | 206.053 | 70.176 | 28.178 | 15.734
$\delta\theta^{\odot}_{\tt gr}~{}[1^{\circ}]$, mas | 1 361 | 622.212 | 233.337 | 77.787 | 15.567 | 4.254 | 1.956

Solar | large $\chi_{1\odot},$ deg
---|---
deflection | $20^{\circ}$ | $40^{\circ}$ | $45^{\circ}$ | $50^{\circ}$ | $60^{\circ}$ | $70^{\circ}$ | $80^{\circ}$ | $90^{\circ}$
$~{}\theta^{\odot}_{\tt gr}$, mas | 23.095 | 11.189 | 9.832 | 8.733 | 7.053 | 5.816 | 4.853 | 4.072
$\delta\theta^{\odot}_{\tt gr}~{}[15^{\circ}]$, mas | 10.180 | 3.366 | 2.778 | 2.341 | 1.746 | 1.372 | 1.122 | 0.948
$\delta\theta^{\odot}_{\tt gr}~{}[1^{\circ}]$, mas | 1.123 | 0.297 | 0.238 | 0.195 | 0.140 | 0.107 | 0.085 | 0.071

Sun-source separation angle, $\chi_{1\odot}$ [deg]

Deflection angle $\log_{10}[\theta^{\odot}_{\tt gr}],~{}[\mu{\rm as}]$

Deflection angle $\log_{10}[\theta^{\odot}_{\tt gr}],~{}[\mu{\rm as}]$

Sun-source separation angle, $\chi_{1\odot}$ [deg]

Figure 4: Solar gravitational deflection of light. On all plots: the upper
thick line is for the absolute astrometric measurements, while the other two
are for the differential astrometry. Thus, the dashed line is for the
observations over field of regard of FoR$\,\,=15^{\circ}$, the lower thick
line is for FoR $=1^{\circ}$.

Figure 4 shows a qualitative presentation of the solar gravitational
deflection. The upper thick line on both plots represents the absolute
astrometric measurements, while the other two are for the differential
astrometry. Thus, the middle dashed line is for the observations over the
maximal field of regard of the instrument FoR$\,\,=15^{\circ}$, the lower
thick line is for FoR $=1^{\circ}$.

One can also account for the post-post-Newtonian (post-PN) terms (e.g.
$\propto G^{2}$) as well as the contributions due to other PPN parameters
Will_book93 . Thus, in the weak gravity field approximation the total
deflection angle $\theta_{\tt gr}$ has an additional contribution due the
post-post-Newtonian terms in the metric tensor. For the crude estimation
purposes this effect could be given by the following expression post-PPN :

$\delta\theta_{\tt post-
PN}=\frac{1}{4}(\gamma+1)^{2}\left(\frac{2GM}{c^{2}d}\right)^{2}\left(\frac{15\pi}{16}-1\right)\left(\frac{1+\cos\chi}{2}\right)^{2}.$
(19)

However, a quick look on the magnitudes of these terms for the solar system’s
bodies suggested that SIM astrometric data will be insensitive to the post-PN
effects. The post-PN effects due to the Sun are the largest among those in the
solar system. However, even for the absolute astrometry with the Sun-grazing
rays the post-PN terms were estimated to be of order $\delta\theta_{\tt post-
PN}^{\odot}=11~{}\mu$as. Note that the SIM solar avoidance angle is
constraining the Sun-source separation angle as $\chi_{1\odot}\geq$ 45∘. The
post-PN effect is inversely proportional to the square of the impact
parameter, thus reducing the effect to $\delta\theta^{\odot}_{\tt post-PN}\leq
4.9$ nanoarcseconds at the edge of the solar avoidance angle. Therefore, the
post-PN effects will not be accessible with SIM.

###  III.2 Gravitational Deflection of Light by Jupiter

Astrometric measurement with SIM would have to account for the light bending
by Jupiter Crosta-Mignard-2006 ; Fomalont-Kopeikin07 . One may obtain the
expression, similar to Eq. (17) for the relativistic deflection of light by
the Jovian gravitational monopole in the following form:

$\theta^{J}_{\tt
gr}=(\gamma+1)~{}\frac{G}{c^{2}}\frac{M_{J}}{r_{J}}~{}\frac{1+\cos\chi_{1J}}{\sin\chi_{1J}}=0.924944\cdot\frac{1+\cos\chi_{1J}}{\sin\chi_{1J}}~{}~{}~{}~{}\mu{\sf
as},$ (20)

with $\chi_{1J}$ being Jupiter-source separation angle as seen by the
interferometer at the distance $r_{J}$ from Jupiter. For the differential
observations one will have expression, similar to that Eq. (18) for the Sun:

$\delta\theta^{J}_{\tt
gr}=(\gamma+1)\frac{G}{c^{2}}~{}\frac{M_{J}}{r_{J}}\frac{\sin\frac{1}{2}(\chi_{2J}-\chi_{1J})}{\sin\frac{1}{2}\chi_{1J}\cdot\sin\frac{1}{2}\chi_{2J}}=0.924944\cdot\frac{\sin\frac{1}{2}(\chi_{2J}-\chi_{1J})}{\sin\frac{1}{2}\chi_{1J}\cdot\sin\frac{1}{2}\chi_{2J}}~{}~{}~{}~{}\mu{\sf
as},$ (21)

where again $\chi_{1J},\chi_{2J}$ are Jupiter-source separation angles for the
primary and secondary stars correspondingly,
$\chi_{2J}=\chi_{1J}+\frac{\pi}{12}$ (and
$\chi_{2J}=\chi_{1J}+\frac{\pi}{180}$ for the narrow angle astrometry). The
largest effect will come when SIM and Jupiter are at the closest distance from
each other $\sim 4.2~{}$AU. Jupiter’s angular dimensions from the Earth’ orbit
for this situation are calculated to be ${\cal R}_{J}=23.24$ arcsec, which
correspond to a deflection angle of 16.419 mas. Results for some $\chi_{1J}$
are given in the Table 6. Note that for the light rays coming perpendicular to
the ecliptic plane the Jovian deflection will be in the range:
$\delta\alpha_{1J}\sim(0.7\\--1.0)~{}\mu$as!

A qualitative behavior of the effect of the gravitational deflection of light
by the Jovian gravity field is plotted in the Figure 5. As in the case of the
solar deflection, the upper thick line on both plots represents the absolute
astrometric measurements, while the other two are for the differential
astrometry (the dashed line is for the observations over FoR$\,\,=15^{\circ}$
and the lower thick line is for FoR $=1^{\circ}$).

Table 6: Jovian gravitational monopole deflection vs. the Jupiter-source sky separation angle $\chi_{1J}$. Jovian | Jupiter-source separation angles $\chi_{1J},~{}~{}$arcsec
---|---
deflection | $23.24^{\prime\prime}$ | $26^{\prime\prime}$ | $30^{\prime\prime}$ | $60^{\prime\prime}$ | $120^{\prime\prime}$ | $180^{\prime\prime}$ | $360^{\prime\prime}$ | $90^{\circ}$
$~{}\theta^{J}_{\tt gr},~{}$mas | 16.419 | 14.676 | 12.719 | 6.360 | 3.180 | 2.120 | 1.060 |  $0.9~{}\mu$as
$\delta\theta^{J}_{\tt gr}[15^{\circ}],~{}$mas | 16.412 | 14.669 | 12.712 | 6.352 | 3.173 | 2.113 | 1.053 |  $0.2~{}\mu$as
$\delta\theta^{J}_{\tt gr}[1^{\circ}],~{}$mas | 16.313 | 14.570 | 12.614 | 6.255 | 3.077 | 2.019 | 0.964 |  $0.0~{}\mu$as

Jupiter-source separation angle, $\chi_{1J}$ [arcsec]

Deflection angle $\log_{10}[\theta^{J}_{\tt gr}],~{}[\mu{\rm as}]$

Deflection angle $\log_{10}[\theta^{J}_{\tt gr}],~{}[\mu{\rm as}]$

Jupiter-source separation angle, $\chi_{1J}$ [deg]

Figure 5: Jovian gravitational deflection of light.

###  III.3 Gravitational Deflection of Light by the Earth

The deflection of light rays by the Earth’s gravity field may also be of
interest. The expressions, describing the relativistic deflection of light by
the Earth’ gravitational monopole are given below:

$\theta^{\oplus}_{\tt
gr}=(\gamma+1)~{}\frac{G}{c^{2}}\frac{M_{\oplus}}{r_{\oplus}}~{}\frac{1+\cos\chi_{1\oplus}}{\sin\chi_{1\oplus}}=0.2446\cdot\frac{1+\cos\chi_{1\oplus}}{\sin\chi_{1\oplus}}~{}~{}~{}~{}\mu{\sf
as},$ (22)

with $\chi_{1\oplus}$ being the Earth-source separation angle as seen by the
interferometer at the distance $r_{\odot}$ from the Earth. Relation for the
differential astrometric measurements was obtained in the form:

$\delta\theta^{\oplus}_{\tt
gr}=(\gamma+1)\frac{G}{c^{2}}~{}\frac{M_{\oplus}}{r_{\oplus}}\frac{\sin\frac{1}{2}(\chi_{2\oplus}-\chi_{1\oplus})}{\sin\frac{1}{2}\chi_{1\oplus}\cdot\sin\frac{1}{2}\chi_{2\oplus}}=0.2446\cdot\frac{\sin\frac{1}{2}(\chi_{2\oplus}-\chi_{1\oplus})}{\sin\frac{1}{2}\chi_{1\oplus}\cdot\sin\frac{1}{2}\chi_{2\oplus}}~{}~{}~{}~{}\mu{\sf
as},$ (23)

where, as before, $\chi_{1\oplus},\chi_{2\oplus}$ are the Earth-source
separation angles for the primary and secondary stars correspondingly,
$\chi_{2\oplus}=\chi_{1\oplus}+\frac{\pi}{12}$ (and
$\chi_{2\oplus}=\chi_{1\oplus}+\frac{\pi}{180}$ for the narrow angle
astrometry). The largest effect will come when SIM and the Earth are at the
closest distance, say at the end of the first half of the first year of the
mission, $r_{\oplus}=0.05$ AU. The Earth’s angular dimensions being measured
from the spacecraft from that distance are calculated to be ${\cal R}^{\tt
SIM}_{\oplus}=175.88401$ arcsec, which correspond to a deflection angle of
573.75 $\mu$as. The summary of the deflection angles for sevral
$\chi_{1\oplus}$ are given in the Table 7.

Table 7: Solar relativistic deflection angle as a function of the Earth-source separation angle.  SIM |  $\chi^{\tt SIM}_{1\oplus},$ arcsec
---|---
mission | 175.88 | 200 | 360 | $1^{\circ}$ | $5^{\circ}$ | $10^{\circ}$ | $15^{\circ}$
$~{}~{}\theta_{1\oplus},~{}\mu$as | 573.8 | 504.7 | 280.3 | 28.0 | 5.6 | 2.8 | 1.9
$\delta\theta_{1\oplus}[15^{\circ}],~{}\mu$as | 571.9 | 502.7 | 278.5 | 26.3 | 4.2 | 1.7 | 1.0
$\delta\theta_{1\oplus}[1^{\circ}],~{}\mu$as | 547.0 | 478.0 | 254.8 | 14.0 | 0.9 | 0.3 | 0.1

At the distance of 0.05 AU from the Earth

Earth-source separation angle, $\chi_{1\oplus}$ [deg]

Deflection angle $\log_{10}[\theta^{\oplus}_{\tt gr}],~{}[\mu{\rm as}]$

Deflection angle $\log_{10}[\theta^{\oplus}_{\tt gr}],~{}[\mu{\rm as}]$

At the distance of 0.5 AU from the Earth

Earth-source separation angle, $\chi_{1\oplus}$ [deg]

Figure 6: Gravitational deflection of light in the proximity of the Earth.

Figure 6 shows the expected variation in the magnitude of the Earth’ gravity
influence as mission progresses. The left plot presents results for the end of
the first half of the year of the mission, when the spacecraft is at the
distance of 0.05 AU from the Earth (the drift rate is 0.1 AU per year). The
plot on the right side is for the end of the 5-th year of the mission, when
SIM is at 0.5 AU from Earth.

###  III.4 Constraints Derived From the Monopole Deflection of Light

While analyzing the solar gravity field’s influence on the future astrometric
observations with SIM, we found several interesting situations, that may
potentially put an additional navigational requirements. In this section we
will consider these situations in a more detailed way.

To measure gravitational deflection of light with an accuracy of
$\Delta\theta_{0}=\Delta k~{}\mu{\rm as}$, one needs to precisely determine
the value of impact parameter of photon’s trajectory with respect to the
deflector. As before, we will present two types of necessary expressions,
namely for absolute and differential observations.

In the case of absolute astrometry we use expression for the deflection angle
$\theta^{B}_{\tt gr}$ from Eq. (10) and present it as $\theta^{B}_{\tt
gr}=({\mu_{B}}/{d})(1+\cos\chi_{B})$, where again $d=r_{B}\sin\chi_{B}$ and
quantity $\mu_{B}=2GM_{B}/c^{2}$ being the relativistic gravitational radius
of the body at question. One may ask a question – what uncertainty in the
knowledge of the impact parameter, $\Delta d$, will result in the astrometric
error of $\Delta\theta_{0}$? The answer is given the following expression

$\Delta
d_{B}=\Delta\theta_{0}\,\,\frac{r^{2}_{B}}{\mu_{B}}\cdot\frac{\sin^{2}\chi_{1B}}{1+\cos\chi_{1B}}.$
(24)

The corresponding result for differential observations may be obtained with
the help of Eq. (13) as:

$\Delta d^{\tt
diff}_{B}=\Delta\theta_{0}\,\,\frac{r^{2}_{B}\sin^{2}\chi_{1B}}{2\mu_{B}}\cdot\Big{[}1+\tan\frac{\chi_{1B}}{2}\cdot\cot\frac{1}{2}(\chi_{2B}-\chi_{1B})\Big{]}.$
(25)

Similarly, the uncertainty in determining the barycentric distance $r_{B}$ is
obtained from Eq. (10) leading to expression:

$\Delta
r_{B}=\Delta\theta_{0}\,\frac{r^{2}_{B}}{\mu_{B}}\cdot\frac{\sin\chi_{1B}}{1+\cos\chi_{1B}}.$
(26)

Note that, when differential observations are concerned, uncertainty in
barycentric position $\Delta r^{\tt diff}_{B}$ does not produce new
constraints significantly different from those derived from Eq. (26). Looking
at the results presented in the Table 8, one may see that for an accuracy of
$\Delta\theta_{0}=1~{}\mu$as our estimates require the knowledge of the solar
impact parameter with the accuracy of $\sim 0.4$ km (grazing rays), that for
Jupiter with the accuracy of $\sim 4~{}$km and other big planets with the
accuracy of about 10 km. Table 9 shows a comparison of these derived
requirements on the barycentric positions of the solar system’s bodies with
the accuracy of their current determination.

Table 8: Required accuracy of barycentric positions and impact parameters for astrometric observations with accuracy of 1 $\mu$as. The Earth is taken at the distance of 0.05 AU from the spacecraft; moon’s position accuracy is for the geocentric frame.  Solar | Required knowledge: grazing rays | Required knowledge: differential astrometry
---|---|---
system’s | Distance, | Impact parameter | Impact param. [15∘] | Impact param. [1∘]
object |  $\sigma_{r_{B}},$ km |  $\sigma_{d_{B}},$ km |  $\sigma_{d_{B}},$ mas |  $\sigma_{r_{B}}$, km |  $\sigma_{r_{B}}$, mas |  $\sigma_{r_{B}}$, km |  $\sigma_{r_{B}}$, mas
Sun | 85.45 | 0.39 | 0.55 | 0.40 | 0.55 | 0.50 | 0.69
Sun at 45∘ | 1.5$\times 10^{4}$ | 7.6$\times 10^{3}$ | 10′′.49 | 3.81$\times 10^{4}$ | 52′′.53 | 4.45$\times 10^{5}$ | 613′′.66
Moon | 2.8$\times 10^{5}$ | 67.14 | 1′′.85 | 67.17 | 1′′.85 | 68.00 | 1′′.88
Mercury | 1.1$\times 10^{6}$ | 29.39 | 66.16 | 29.41 | 66.16 | 29.45 | 66.25
Venus | 8.4$\times 10^{4}$ | 12.18 | 61.00 | 12.28 | 61.20 | 12.38 | 62.70
Earth-Moon | 1.3$\times 10^{4}$ | 11.14 | 306.55 | 11.15 | 307.47 | 11.66 | 321.54
Mars | 6.8$\times 10^{5}$ | 29.29 | 77.11 | 29.30 | 77.14 | 29.37 | 77.33
Jupiter | 3.8$\times 10^{4}$ | 4.31 | 1.42 | 4.32 | 1.42 | 4.34 | 1.42
Jupiter at 30′′ | 4.9$\times 10^{4}$ | 7.14 | 2.34 | 7.20 | 2.36 | 7.25 | 2.38
Saturn | 2.2$\times 10^{5}$ | 10.32 | 1.66 | 10.34 | 1.66 | 10.36 | 1.66
Uranus | 1.2$\times 10^{6}$ | 11.27 | 0.86 | 11.28 | 0.86 | 11.29 | 0.86
Neptune | 1.7$\times 10^{6}$ | 10.04 | 0.47 | 10.04 | 0.47 | 10.04 | 0.47
Pluto | 2.0$\times 10^{9}$ | 1 133.92 | 40.7 | 1 133.93 | 40.7 | 1133.96 | 40.7

One may see that the present accuracy of knowledge of the inner planets’
positions from the Table 9 is given by the radio observations and it is even
better than the level of relativity requirements given in the Table 8.
However, the positional accuracy for the outer planets is below the required
level. The SIM observation program should include the astrometric studies of
the outer planets in order to minimize the errors in their positional accuracy
determination. Thus, in order to get the radial uncertainty in Pluto’s
ephemeris with accuracy below 1000 km, it is necessary only 4 measurements of
Pluto’s position, taken sometime within a week of the stationary points,
spread over 3 years. Each measurement could be taken with an accuracy of about
$200~{}\mu$as, as suggested by Standish95 . Additionally, one will have to
significantly lean on the radio observations in order to conduct the reduction
of the optical data with an accuracy of a few $\mu$as. For this reason one
will have to use the precise catalog of the radio-sources and to study the
problem of the radio and optical reference frame ties Standish95 ;
Standish_etal95 ; Folkner94 ; Ma-1998 .

The estimates, presented here were given for static gravitational field.
Analysis of a real experimental situation should consider a non-static
gravitational environment of the solar system and should include the
description of light propagation in a different reference frames involved in
the experiment Kopeikin-Schafer-1999 . Additionally, the observations will be
affected by the relativistic orbital dynamics of the spacecraft Turyshev98 .

Table 9: The best known accuracies of barycentric positions and masses for the solar system’s objects Yoder95 ; Pitjeva-2005 ; Folkner-Williams-Boggs-2008 .  Solar | Knowledge of barycentric position | Knowledge of
---|---|---
system’s | Best known, | Method used | planetary masses,
object |  $\sigma_{r_{B}},$ km |  $\sigma_{r_{B}},$ mas | for determination | $\Delta M_{B}/M_{B}$
Sun | 362/725 | $0^{\prime\prime}.5/1^{\prime\prime}.0$ | Optical meridian transits | 3.77$\times 10^{-10}$
Moon | 27 cm | 7.4 $\mu$as | LLR, 1995 | 1.02$\times 10^{-6}$
Mercury | 1 | 2.25 | Radar ranging | 4.13$\times 10^{-5}$
Venus | 1 | 4.98 | Radar ranging | 1.23$\times 10^{-7}$
Earth | 1 | 27.58 | Radar ranging | TBD$\times 10^{-6}$
Mars | 1 | 2.63 | Radar ranging | 2.33$\times 10^{-6}$
Jupiter | 30 | 9.84 | Radar ranging | 7.89$\times 10^{-7}$
Saturn | 350 | 56.24 | Optical astrometry | 2.64$\times 10^{-6}$
Uranus | 750 | 57.00 | Optical astrometry | 3.97$\times 10^{-6}$
Neptune | 3 000 | 141.67 | Optical astrometry | 2.19$\times 10^{-6}$
Pluto/Charon | 20 000 | 717.40 | Photographic astrometry | 0.014

##  IV Deflection of light by higher gravity multipoles

In order to carry out a complete analysis of the relativistic light deflection
one should account for other possible terms in the expansion (1) that may
potentially contribute to this effect. These terms are due to non-sphericity
and non-staticity of the body’s gravity field Kopeikin-1997 . Here we will
consider several of them, namely those due to higher gravitational multipoles
of the celestial bodies.

###  IV.1 Gravitational quadrupole deflection of light

Although a complete three-dimensional deflection of light must be considered
for a real experiment, for the purposes of this paper, we consider only two
dimensional case. In this case, quadrupole term may be given as Turyshev98 ;
Klioner-1991 ; Kopeikin-Makarov-2007 :

$\theta_{J_{2}}=\frac{1}{2}(\gamma+1)\frac{4GM_{B}}{c^{2}{\cal
R}_{B}}J_{2B}\Big{(}1-s_{z}^{2}-2u_{z}^{\hskip 2.0pt2}\Big{)}\left({{\cal
R}_{B}\over d}\right)^{3},$ (27)

where $J_{2}$ is the second zonal harmonic of the body under question,
$\vec{s}=(s_{x},s_{y},s_{z})$ is the unit vector in the direction of the light
ray propagation and vector $\vec{d}=d(u_{x},u_{y},u_{z})$ is the impact
parameter. A similar expression may also be obtained for differential
observations. For estimation purposes, this formula may be given as follows
($d=r_{B}\sin\chi_{B}$):

$\delta\theta_{J_{2}}\approx\frac{4\mu_{B}J_{2B}{\cal
R}^{2}_{B}}{r^{3}_{B}}~{}\Big{[}\,\frac{1}{\sin^{3}\chi_{1B}}-\,\frac{1}{\sin^{3}\chi_{2B}}\Big{]}.$
(28)

The corresponding effects for the deflection of light by the quadrupole mass
moments within the planets of the solar system are given in the Table 11. The
effect depends on a number of different instantaneous geometric parameters
defining the mutual orientation of the vector of the light propagation,
position of the planet in orbit, the orientation of the axes defining it’s
figure, etc. A mission-independent modeling in the static gravitational regime
has been done Crosta-Mignard-2006 ; the modeling of the dynamic regime was
presented in Kopeikin-Makarov-2007 . An effort to develop a SIM-specfic model
is well justified.

The quadruple deflection of light depends on the third power of the inverse
impact parameter with respect to the deflecting body, Eq. (27). SIM will
measure this effect directly for many celestial bodies. At the expected level
of accuracy the knowledge of jovian atmosphere, the magnetic field
fluctuations, etc., may contribute to the errors in the experiment
TreuhaftLowe91 . A detailed study of these effects is given in Kopeikin-
Fomalont-02 .

Table 10: Higher gravitational coefficients for solar system bodies (http://nssdc.gsfc.nasa.gov/planetary/factsheet/). Solar system’s | $J^{B}_{2},$ | $J^{B}_{4},$ | $J^{B}_{6},$
---|---|---|---
object | $\times~{}10^{-6}$ | $\times~{}10^{-6}$ | $\times~{}10^{-6}$
Sun | $0.17\pm 0.017$ | — | —
Sun at 45∘ | — | — | —
Moon | 202.2 | $-0.1$ | —
Mercury | 60. | — | —
Venus | 4.5 | $-2.1$ | —
Earth | 1 082.6 | $-1.6$ | 0.5
Mars | 1 960.45 | — | —
Jupiter | 14 738$\pm$1 |  $-$587$\pm$5 | 34$\pm$50
Saturn | 16 298$\pm 50$ |  $-$915$\pm$80 | $103.0$
Uranus | 3 343.43 | — | —
Neptune | 3 411. | — | —
Pluto | — | — | —
Table 11: Relativistic deflection of light by the planetary quadrupole mass moments for solar system bodies. Solar system’ | $J^{B}_{2},$ | $\theta^{B}_{J_{2}},$ | $d^{\tt crit}_{{J_{2}}}$ | $\delta\theta^{B}_{J_{2}}[15^{\circ}],$ | $\delta\theta^{B}_{J_{2}}[1^{\circ}],$
---|---|---|---|---|---
object | $\times~{}10^{-6}$ |  $\mu$as |  |  $\mu$as |  $\mu$as
Sun | $0.17\pm 0.017$ | 0.3 | — | 0.3 | 0.3
Moon | 202.2 | 2$\times 10^{-2}$ | — | 2$\times 10^{-2}$ | $2\times 10^{-2}$
Mercury | 60. | 5$\times 10^{-3}$ | — | — | —
Venus | 4.5 | 2$\times 10^{-3}$ | — | — | —
Earth | 1 082.6 | 0.6 | — | 0.6 | 0.6
Mars | 1 960.45 | 0.2 | — | 0.2 | 0.2
Jupiter | 14 738$\pm$1 | 242.0 | 98′′.12 – 144′′.81 | 242.0 | 242.0
|  |  | 6.23 ${\cal R}_{J}$ |  |
Saturn | 16 298$\pm$ 50 | 94.6 | 35′′.62 – 43′′.93 | 94.6 | 94.6
|  |  | 4.56 ${\cal R}_{S}$ |  |
Uranus | 3 343.43 | 7.3 | 3′′.25 – 3′′.61 | 7.3 | 7.3
|  |  | 1.94 ${\cal R}_{U}$ |  |
Neptune | 3 411. | 8.5 | 2′′.23 – 2′′.42 | 8.5 | 8.5
|  |  | 2.04 ${\cal R}_{N}$ |  |
Pluto | — | — | — | — | —

Table 11 presents estimates of the magnitudes of the relativistic deflection
of light by the planetary quadrupole mass moments for solar system bodies.
Based on the sizes of these effects, one would have to account for the
quadrupole component of the gravity fields when observations will be conducted
in the vicinity of the outer planets. In addition, the influence of the higher
harmonic may be also of interest. Thus, Table 11 shows the estimates of some
higher gravitational multipole moments of Jupiter and Saturn. We will discuss
the deflection by the $J_{2}$ and $J_{4}$ coefficients of the jovian gravity
field in terms of Jupiter-source separation angle $\chi_{1J}$. An expression,
similar to that of Eq. (20) for the monopole deflection, may be given as:

$\theta^{\tt max}_{J_{2}}={3.46058\times
10^{-10}}~{}\frac{1}{\sin^{3}\chi_{1J}}~{}~{}\mu{\sf as}.$ (29)

Jupiter’s angular dimensions from the Earth are calculated to be ${\cal
R}_{J}=23.24~{}$arcsec, which correspond to a deflection angle of
$242~{}\mu$as. The deflection on the multipoles for some $\chi_{1J}$ is given
in the Table 13.

Table 12: Deflection of light by the Jovian higher gravitational coefficients.

Jovian |  $\chi_{1J}$, arcsec
---|---
deflection | $23^{\prime\prime}.24$ | 26′′ | 30′′ | 35′′ | 40′′ | 50′′ | 120′′
$\theta^{J}_{J_{2}},~{}\mu$as | 242 | 173 | 112 | 71 | 47 | 24 | 1.8
$\delta\theta^{J}_{J_{2}}[15^{\circ}],~{}\mu$as | 242 | 173 | 112 | 71 | 47 | 24 | 1.8
$\theta^{J}_{J_{4}},~{}\mu$as | 9.6 | 5.5 | 2.7 | 1.3 | 0.6 | 0.2 | 0.0

Table 13: Deflection of light by the Saturnian higher gravitational
coefficients.

Saturnian |  $\chi_{1S}$, arcsec
---|---
deflection | $9^{\prime\prime}.64$ | 12′′ | 15′′ | 20′′ | 25′′ | 30′′ | 35′′
$\theta^{S}_{J_{2}},~{}\mu$as | 94.7 | 49.1 | 25.1 | 10.6 | 5.4 | 3.1 | 2
$\delta\theta^{S}_{J_{2}}[15^{\circ}],~{}\mu$as | 94.7 | 49.1 | 25.1 | 10.6 | 5.4 | 3.1 | 2
$\theta^{S}_{J_{4}},~{}\mu$as | 5.3 | 1.8 | 0.6 | 0.1 | — | — | —

Similar studies are important for Saturn (see Kopeikin-Makarov-2007  for
details). In terms of the Saturn-source separation angle $\chi_{1S}$ the
saturnian quadrupole deflection mat be estimated with the help of the
following expression:

$\theta^{\tt max}_{J_{2}}=9.66338\times
10^{-12}~{}\frac{1}{\sin^{3}\chi_{1S}}~{}~{}\mu{\sf as}.$ (30)

The Saturn’s angular dimensions from the Earth’ orbit are calculated to be
${\cal R}_{S}=9.64~{}$arcsec, which correspond to a deflection angle of
$94.7~{}\mu$as. The corresponding estimates for the deflection angles are
presented in the Table 13.

As a result, for astronomical observations with accuracy of about $1~{}\mu{\rm
as}$, one will have to account for the quadrupole gravitational fields of the
Sun, Jupiter, Saturn, Neptune, and Uranus. In addition, the influence of the
higher harmonics may be of interest. For example some of the moments for
Jupiter and Saturn are given in the Table 10.

Higher multipoles may also influence the astrometric observations taken close
to these planets. Thus, for both Jupiter and Saturn the rays, grazing their
surface, will be deflected by the fourth zonal harmonic $J_{4}$ as follows:
$\delta\theta^{J}_{J_{4}}\approx 9.6~{}\mu$as,
$\delta\theta^{S}_{J_{4}}\approx 5.3~{}\mu$as. In addition, the contribution
of the $J_{6}$ for Jupiter and Saturn will deflect the grazing rays on the
angles $\delta\theta^{J}_{J_{6}}\approx 0.8~{}\mu$as,
$\delta\theta^{S}_{J_{6}}\approx 0.6~{}\mu$as. The contribution of $J_{4}$ is
decreasing with the distance from the body as $d^{-5}$ and contribution of
$J_{6}$ as $d^{-7}$. As a result the deflection angle will be less then
$1~{}\mu$as when $\hskip 2.0ptd>1.6~{}{\cal R}$, where ${\cal R}$ is the
radius of the planet.

Using Eq. (27), one can derive the expression for the critical distance
$d^{\tt crit}_{J_{2}}$ for the astrometric observations in the regime of
quadrupole deflection of light with accuracy of $\Delta\theta_{0}=\Delta
k~{}\mu$as. Indeed, approximating this equation as
$\theta_{J_{2}}\simeq({4\mu_{B}}/{d^{3}})J_{2}{\cal R}^{2}$ and solving it for
$d$ one obtaines the following result:

$d^{\tt crit}_{J_{2}}={\cal R}_{B}\Big{[}\frac{4\mu_{B}}{{\cal
R}_{B}}\frac{J^{B}_{2}}{\Delta\theta_{0}}\Big{]}^{\frac{1}{3}}.$ (31)

The critical distances for the relativistic quadrupole deflection of light by
the solar system’s bodies for the case of $\Delta k=1$ presented in the Table
11.

###  IV.2 Gravito-Magnetic Deflection of Light

Besides the gravitational deflection of light by the monopole and the
quadrupole components of the static gravity field of the bodies, the light ray
trajectories will also be affected by the non-static contributions from this
field. It is easy to demonstrate that a rotational motion of a gravitating
body contributes to the total curvature of the space-time generated by this
same body. This contribution produces an additional deflection of light rays
Klioner-1991 ; Ciufolini-etal-2003 ; Kopeikin-Mashhoon-2003  on the angle

$\delta\theta_{\vec{\cal
S}}=\frac{1}{2}(\gamma+1)\frac{4G}{c^{3}d^{3}}\vec{\cal
S}(\vec{s}\cdot\vec{d}),$ (32)

where ${\vec{\cal S}}$ is the body’s angular momentum.

The most significant contributions of gravito-magnetic deflection of light by
the bodies of the solar system are the following ones: i) the solar deflection
amounts to $\delta\theta^{\odot}_{\vec{\cal S}}=\pm(0.7~{}-1.3)\mu$as (the
first term listed is for a uniformly rotating Sun; the second is for the
Dicke’s model Dicke-1974 ); ii) jovian rotation contributes
$\delta\theta^{J}_{\vec{\cal S}}=\pm 0.2~{}\mu$as; and iii) Saturn’s rotation
$\delta\theta^{Sa}_{\vec{\cal S}}=\pm 0.04~{}\mu$as. Thus, depending on the
model for the solar interior, solar rotation may produce a noticeable
contribution for the grazing rays. The estimates of magnitude of deflection of
light ray’s trajectory, caused by the rotation of gravitating bodies
demonstrate that for precision of observations of $1~{}\mu$as it is sufficient
to account for influence of the Sun and Jupiter only.

The relativistic gravito-magnetic deflection of light has never been directly
tested before. However, because of the smallness of the magnitudes of
corresponding effects in the solar system and SIM’s operational mode that
limits the viewing angle for a sources as $\chi_{1\odot}\geq 45^{\circ}$, SIM
will not be sensitive to this effect.

##  V Astrophysics Investigations with SIM

###  V.1 Astrometric Test of General Relativity

The Eddington parameter $\gamma$ in Eq. (II.1), whose value in general
relativity is unity, is perhaps the most fundamental PPN parameter Will-
lrr-2006-3 , in that $\frac{1}{2}(1-\gamma)$ is a measure, for example, of the
fractional strength of the scalar gravity interaction in scalar-tensor
theories of gravity Damour-Nordtvedt-1993 . Currently, the most precise value
for this parameter, $\gamma-1=(2.1\pm 2.3)\times 10^{-5}$, was obtained using
radio-metric tracking data received from the Cassini spacecraft Bertotti-Iess-
Tortora-2003  during a solar conjunction experiment. This accuracy approaches
the region where multiple tensor-scalar gravity models, consistent with the
recent cosmological observations Spergel-etal-2007 , predict a lower bound for
the present value of this parameter at the level of $(1-\gamma)\sim
10^{-6}-10^{-7}$ Damour-Nordtvedt-1993 . Therefore, improving the measurement
of this parameter would provide the crucial information separating modern
scalar-tensor theories of gravity from general relativity, probe possible ways
for gravity quantization, and test modern theories of cosmological evolution
Turyshev-etal-2007 ; Turyshev-2008 .

The reasons above led to a number of specific space experiments dedicated to
measurement of the parameter $\gamma$ with a precision better than $10^{-5}$
to $10^{-6}$ Turyshev-etal-2007 ; Turyshev-2008 . Note that SIM will operate
at this level of accuracy and, therefore, the Eddington’s parameter $\gamma$
will have to be included into the future SIM’s astrometric model and the
corresponding data analysis.

####  V.1.1 Solar Gravity Field as a Deflector

To model the astrometric data to the nominal measurement accuracy will require
including the effect of general relativity on the propagation of light. In the
PPN framework, the parameter $\gamma$ would be part of this model and could be
estimated in global solutions. The astrometric residuals may be tested for any
discrepancies with the prescriptions of general relativity. To address this
problem in a more detailed way, one will have to use the astrometric model for
the instrument including the information about it’s position in the solar
system, it’s attitude orientation in the proper reference frame, the time
history of different pointings and their durations, etc. This information then
should be folded into the parameter estimation program that will use a model
based on the expression, similar to that given by Eq. (12).

To estimate the expected accuracy of the parameter $\gamma$, we use Eq. (13)
to assume that the single astrometric measurement may be able to determine
this parameter with accuracy:

$\Delta\gamma=\Delta\theta_{0}~{}\frac{r^{\tt
SIM}_{\odot}}{\mu_{\odot}}~{}\frac{\sin\frac{1}{2}\chi_{1\odot}\,\sin\frac{1}{2}\chi_{2\odot}}{\sin\frac{1}{2}(\chi_{2\odot}-\chi_{1\odot})},$
(33)

where $\Delta\theta_{0}$ is the astrometric error of the measurement.

The relativity test will be enhanced by scheduling measurements of stars as
close to the Sun as possible. Although SIM will never be able observe closer
to the Sun than $45^{\circ}$, it will allow for an accurate determination of
this PPN parameter. For the accuracy of $\Delta\theta_{0}=1~{}\mu$as at the
rim of the solar avoidance angle of $\chi_{1\odot}=45^{\circ}$, one could
determine this parameter with an accuracy $\sigma_{\gamma}\sim 7.19\times
10^{-4}$ in a single measurement. Assuming Gaussian error distribution, the
accuracy of this experiment will improve as $1/\sqrt{N}$, where $N$ is the
number of independent observations. Therefore, by performing differential
astrometric measurements with an accuracy of $\Delta\theta_{0}=1~{}\mu$as over
the instrument’s FoR=15∘, at the end of the mission (with $N\sim 10000$
observations) SIM may reach the accuracy of $\sigma_{\gamma}\sim 7.2\times
10^{-6}$ in astrometric test of general relativity in the solar gravity field.
SIM will provide this precision as a by-product of its astrometric program,
thus allowing for a factor of 3 improvement of the currently best Cassini’s
2003 result Bertotti-Iess-Tortora-2003 . Such a measurement improves the
accuracy of the search for cosmologically relevant scalar-tensor theories of
gravity by looking for a remnant scalar field in today’s solar system.

####  V.1.2 GR Test in the Jovian and Earth’ Gravity Fields

One could also perform a relativity experiment with Jupiter and the Earth. In
fact, for the proposed SIM’s observing mode, the accuracy of determining of
the parameter $\gamma$ may be even better than that achievable with the Sun.
Indeed, with the same assumptions as above, one may achieve a single
measurement the accuracy of $|\gamma-1|\sim 4.0\times 10^{-4}$ determined via
deflection of light by Jupiter. As the astrometric observations in the
Jupiter’s vicinity will require careful planning thereby minimizing the number
of possible independent observations. As a result, the PPN parameter $\gamma$
may be obtained with accuracy of about $\sigma_{\gamma}\sim 1.3\times 10^{-5}$
with astrometric experiments in Jupiter’s gravity field (note that only
$N\sim$ 1000 needed).

Lastly, let us mention that the experiments conducted in the Earth’s gravity
field, could also determine this parameter to an accuracy $|\gamma-1|\sim
8.9\times 10^{-3}$ in a single measurement (which in return extends the
measurement of the gravitational bending of light to a different mass and
distances scale, as shown by Gould93 ). One may expect a large statistics
gained from both the astrometric observations and the telecommunications with
the spacecraft. This, in return, will significantly enhance the overall
solution for $\gamma$ obtained in the Earth’ gravitational environment.

###  V.2 Solar Acceleration Towards the Galactic Center

The Sun’s absolute velocity with respect to a cosmological reference frame was
measured photometrically: it shown up as the dipole anisotropy of the cosmic
microwave background Spergel-etal-2007 . The Sun’s absolute acceleration with
respect to galactic frame can be measured astrometrically: it will show up as
a dipole vector harmonic in the global pattern of proper motion of quasars.

The aberration due to the solar system’s galactocentric motion will not be
observable because its main contribution is static. However, the rate of this
aberration will produce an apparent proper motion for the observed sources
Kopeikin-Makarov-2006 . Indeed, the solar system’s orbital velocity around the
galactic center causes an aberrational affect of the order of 2.5 arcmin. All
measured star and quasar positions are shifted towards the point on the sky
having galactic coordinates $l=90^{\circ},~{}b=0^{\circ}$. For an arbitrary
point on the sky the size of the effect is $2.5~{}\sin\eta$ arcmin, where
$\eta$ is the angular distance to the point $l=90^{\circ},~{}b=0^{\circ}$. The
acceleration of the solar system towards the galactic center causes this
aberrational effect to change slowly. This leads to a slow change of the
apparent position of distant celestial objects, i.e. to an apparent proper
motion.

Let us assume a solar velocity of $220$ km/sec and a distance of 8.5 kpc to
the galactic center. The orbital period of the Sun is then 250 million years,
and the galactocentric acceleration takes a value of about $1.75\times
10^{-13}$ km/sec2. Expressed in a more useful units it is 5.5 mm/s/yr. A
change in velocity by 5.5 mm/sec causes a change in aberration of the order of
4 $\mu$as. The apparent proper motion of a celestial object caused by this
effect always points towards the direction of the galactic center. Its size is
$~{}4\sin\eta~{}\mu$as/yr, where $\eta$ is now the angular distance between
the object and the galactic center.

The above holds in principle for quasars, for which it can be assumed that the
intrinsic proper motions (i.e. those caused by real transverse motions) are
negligible. A proper motion of $4~{}\mu$as/yr corresponds to a transverse
velocity of $2\times 10^{4}$ km/sec at $z=0.3$ for $H_{0}$=100 km/sec/Mpc, and
to $4\times 10^{4}$ km/sec for $H_{0}=50$ km/sec/Mpc. Thus, all quasars will
exhibit a distance-independent steering motion towards the galactic center.
Within the Galaxy, on the other hand, the effect is drowned in the local
kinematics: at 10 pc it corresponds to only 200 m/sec.

However, for a differential astrometry with SIM this effect will have to be
scaled down to account for the size of the field of regard Turyshev98 , namely
$2\sin\frac{\tt FoR}{2}=2\sin\frac{\pi}{24}=0.261$. This fact is reducing the
total effect of the galactocentric acceleration to only $~{}\sim
1\sin\eta~{}\mu$as/yr and, thus, it makes the detection of the solar system’s
galactocentric acceleration with SIM to be a quite problematic issue.

## Discussion

General relativistic deflection of light produces a significant contribution
to the future astrometric observations with accuracy of about a few $\mu$as.
In this paper we addressed the problem of light propagation on the
gravitational field of the solar system. It was shown that for high accuracy
observations it is necessary to correct for the post-Newtonian deflection of
light by the monopole components of gravitational fields of a large number of
celestial bodies in the solar system, namely the Sun and the nine planets,
together with the planetary satellites and the largest asteroids (important
only if observations are conducted in their close proximity). The most
important fact is that the gravitational presence of the Sun, Jupiter and the
Earth should be always taken into account, independently on the positions of
these bodies relative to the interferometer. It is worth noting that the post-
post Newtonian effects due to the solar gravity are unlikely to be accessible
with SIM. This effect as well as the effect of gravitational deflection of
light caused by the mass quadrupole term of the Sun are negligible at the
level of expected accuracy. However, deflection of light by some planetary
quadrupoles may have a big impact on the astrometric accuracy. Thus, the
higher gravitational multipoles should be taken into account when observations
are conducted in the close proximity of two bodies of the solar system,
notably Jupiter and Saturn.

We emphasized the need of development of a general relativistic model for SIM
observables to enable the mission to improve the current astrometric accuracy
by a factor of over 1000. This model would have to account for a number of
dynamical effects both external to the spacecraft (e.g., motion with respect
to the solar system barycentric reference frame, effects of time-varying
gravitational field in the solar system (due to planetary motion nd rotation)
on light propagation, various interplanetary media effects, etc.) and internal
to the spacecraft (e.g., systematic effects introduced by the spacecraft
itself). Some of this work has already begun in the context of the development
relativistic reference frames for the need of future high-precision
observations Kopeikin-Makarov-2007 . However, a lot more efforts is needed;
this paper intends to motivate initiation of such a work in the near future.

## Acknowledgments

The reported research has been done at the Jet Propulsion Laboratory,
California Institute of Technology, which is under contract to the National
Aeronautic and Space Administration.

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