---
abstract: 'We present results from a covariance study for the proposed Laser Astrometric Test of Relativity (LATOR) mission. This mission would send two laser-transmitter spacecraft behind the Sun and measure the relative gravitational light bending of their signals using a hundred-meter-baseline optical interferometer to be constructed on the International Space Station. We assume that each spacecraft is equipped with a $ < 1.9 \times 10^{-13} \mathrm{m} \mathrm{s}^2 \mathrm{Hz}^{-1/2} $ drag-free system and assume approximately one year of data. We conclude that the observations allow a simultaneous determination of the orbit parameters of the spacecraft and of the Parametrized Post-Newtonian (PPN) parameter $\gamma$ with an uncertainty of $2.7 \times 10^{-9}$. We also find a $5 \times 10^{-9}$ determination of the solar quadrupole moment, $J_2$, as well as the first measurement of the second-order post-PPN parameter $\delta$ to an accuracy of about $10^{-3}$.'
author:
- 'Joseph E. Plowman'
- 'Ronald W. Hellings'
bibliography:
- 'LATOR\_covariance\_paper\_revtex.bib'
date: 'May 26, 2005'
title: LATOR Covariance Analysis
---

Introduction
============

In this paper, we present results of a covariance study for the proposed Laser Astrometric Test of Relativity (LATOR) mission (see @0264-9381-21-12-001). The goal of the LATOR Mission is to effect a precise measurement of the gravitational bending of light for laser signals passing near the Sun and thereby determine several Parametrized Post-Newtonian (PPN) parameters of relativistic gravity with an accuracy many orders of magnitude better than present best estimates.

The LATOR instrument consists of two Sun-orbiting laser transmitter spacecraft in eccentric ecliptic orbits with a 3/2 period resonance with the Earth’s orbit. The two spacecraft are separated by about 1 degree, as seen from the Earth. The angular position of each spacecraft is measured using a 100m-baseline optical interferometer, proposed for the Earth-orbiting International Space Station (ISS). Figure \[latorf1\] shows the LATOR mission geometry. The coherent laser signals from each spacecraft will allow the position of each to be determined with an accuracy of 0.1 picoradians (20 nanoarcseconds), relative to the reference frame defined by the interferometer baseline. The simultaneous measurement of two spacecraft enables a relative angle to be measured, eliminating the need to tie the interferometer frame to inertial space.

![\[latorf1\] Diagram of LATOR geometry: The transmitting spacecraft are on far side of the Sun from Earth; the receiving spacecraft are in low Earth orbit aboard the ISS](LATOR_diagram.eps){width="6in"}

As the lines-of-sight from the spacecraft to the Earth pass close to the Sun, the gravitational bending of light will affect the inner spacecraft’s apparent position more than it does the outer spacecraft, causing the inter-spacecraft angle, as seen from Earth, to differ from its value in the absence of gravitational effects. Measurements of this inter-spacecraft angle may thus detect the gravitational bending of light and verify gravitational theories that predict it. Many measurements of this angle will be made as, over the course of a year, the spacecraft will make three such passes behind the Sun, giving detailed information about how the bending of light changes depending on which part of the solar gravitational field the light passes through (see figure \[orbitplot\]). For more information on the design of LATOR, see @Turyshev:2004vn [@Turyshev:2004ga].

![\[orbitplot\] Plot of impact parameter vs. time for orbits of spacecraft 1 (crosses) and spacecraft 2 (diamonds), with 1 day sampling interval. Time axis is in days, vertical (impact parameter) axis is in units of $R_{SUN}$.](orbits.eps){width="6in"}

One of the problems, of course, in measuring the gravitational deflection of the laser signals from LATOR is to know what the apparent angle between the two spacecraft would be without the gravitational bending present. Two methods have been proposed to provide this knowledge. One is to have a microwave (or optical) signal sent between the two spacecraft, determining their separation to an accuracy of $\sim 1$ cm. The other is to equip each LATOR spacecraft with a drag-free system, so that the orbit of each is very close to a purely gravitational and noise-free trajectory. The spacecraft-to-spacecraft distance can then be inferred by accurate tracking and orbit-modeling over the time of the mission. As we will discuss in section \[ipsec\], the spacecraft-to-spacecraft link leaves one important problem unsolved, that of determining the impact parameter. However, as we show by means of this covariance study, the drag-free system allows orbit modeling using the primary LATOR data to simultaneously solve the spacecraft-to-spacecraft distance and the impact parameter problems.

LATOR Science Goals
===================

As was shown in the early 1970’s, most theories of gravity are metric theories in which the dynamics of neutral matter are determined by a metric, which is itself determined by the neighboring mass distribution. @1972ApJ...177..757W developed an expansion of the metric that has a tunable set of numerical parameters which allow it to describe a very broad range of metric theories of gravity. This Parameterized Post-Newtonian (PPN) expansion is useful for experimental tests of gravity because the task of fitting the model to the data is carried out by simply adjusting the numerical values of these tunable parameters so that the model best fits the data. For the LATOR experiment, the version of the PPN metric required is: $$\begin{aligned}
 %\label{eq:g001}%
g_{00} & = & -1+2 \frac{GM}{r}(1-J_2\frac{R^2}{r^2}\frac{3 \cos^2 \theta - 1}{2})-2\beta \frac{G^{2}M^{2}}{r^{2}} \label{eq:g0k1}\\
%\end{equation}%
%\begin{equation} \label{eq:g0k1}%
%g_{0k} & = & \frac{1}{2}(4\gamma +3+\alpha_{1}+\alpha_{2})\frac{GMv^{k}}{r}+\frac{1}{2}(1+\alpha_{2})\frac{G M x^{k}}{r^{3}}(\vec{r} \cdot \vec{v}) \\%
%\end{equation}%
%\begin{equation} \label{eq:glk1}%
g_{lm} & = & \delta_{lm}(1+2\gamma \frac{GM}{r}[1-J_2\frac{R^2}{r^2}\frac{3 \cos^2 \theta - 1}{2}]+\frac{3}{2} \delta \frac{G^{2}M^{2}}{r^{2}}),\label{eq:glk1}\end{aligned}$$ where $M$ is the mass of the Sun, $R$ is the solar radius, $r$ and $\theta$ are polar coordinates, $J_2$ is the solar quadrupole moment, and $\gamma$, $\beta$ and $\delta$ are PPN parameters. While the full PPN metric contains other terms as well, this version contains the terms that contribute significantly to the bending of light that is observable by LATOR. There is a frame-dragging term that scales with the solar rotational angular momentum, $J_z$, and which should have an effect similar in size to that of other terms observable by LATOR (see @1983PhRvD..28.3007R ). This term is not included in our model, but should not be difficult to add at a later date. Table \[ppnsig\] contains a description of the effects of each parameter in the above metric, along with its current best estimate and theoretical value in Einstein’s General Relativity (GR).

   Parameter  Significance                                         Current best estimate
  ----------- ---------------------------------------------------- -----------------------------------------------------------------------------
   $\gamma$   Space-time curvature per unit rest mass.             $ 1 + (2.1 \pm 2.3)\times 10^{-5} $[@Bertotti:2003rm]. Equal to one in GR
    $\beta$   Space curvature per unit gravitational self-energy   $1+(0.9\pm 1.1)\times 10^{-4}$[@2004PhRvL..93z1101W]. Equal to one in GR
   $\delta$   Time curvature per unit gravitational self-energy    Unmeasured. Equal to one in GR
     $J_2$    Solar quadrupole moment parameter.                   Never measured directly. Estimated from solar models to be about $10^{-7}$.

  : \[ppnsig\] Significance and best estimates for parameters in PPN metric

The accuracy of an interferometer angular measurement of gravitational deflection is related to the accuracy of the measurement of the differential time-of-arrival of a phase front in the two detectors of the interferometer. The relation is $$\Delta\theta=\frac {c\Delta\tau} {L}$$ where $\Delta\theta$ is the accuracy of the angular measurement in radians, $\Delta\tau$ is the accuracy of the timing measurement in seconds, and $L$ is the interferometer baseline in meters. LATOR’s 20 nanoarcsecond (or $10^{-13}$ radian) angular accuracy thus corresponds to $3.33\times10^{-20}$ s (or about 20 $\mu$cycles) accuracy in timing measurement at each element of the interferometer. To second order in $GM/r$, the time delay derived from the metric of eqs. \[eq:g0k1\] and \[eq:glk1\] may be written (see @1983PhRvD..28.3007R) $$\begin{aligned}
\frac{\tau_{TR}}{R} & = &\frac{x_R-x_T}{R}(1-\frac{GM}{R}\frac{R}{r_T}+\frac{1}{2}(1+\gamma^2)\frac{(GM)^2}{R^2}\frac{R^2}{y_T^2}((1-\frac{x_T}{r_T})^2+2\frac{x_R x_T-y_T^2}{r_R r_T}) \nonumber\\
& & {}+\frac{1}{2}(2\beta-1)\frac{(GM)^2}{R^2}\frac{R^2}{r_T^2})+((1+\gamma)\frac{GM}{R}(1-\frac{GM}{R}\frac{R}{r_T}))\ln\frac{x_R+r_R}{x_T+r_T} \nonumber\\
& & {}+(2(1+\gamma)-\beta+\frac{3}{4}\delta )\frac{(GM)^2}{R^2}\frac{R}{y_T}(\arctan\frac{x_R}{y_T}-\arctan\frac{x_T}{y_T})+ \nonumber\\
& & {}+(1+\gamma)^2\frac{(GM)^2}{R^2}\frac{R}{y_T}\frac{r_T}{y_T}(1-\frac{r_R}{r_T})(\frac{x_T}{r_T}+\frac{x_R}{r_R}-1)+\frac{1}{2}(1+\gamma) J_2 \frac{GM}{R}\frac{R^2}{y_T^2}\label{tofeq}\end{aligned}$$ Here, it is assumed that the $x$ axis is tangent to the photon trajectory at the transmitter, the origin is at the position of the source mass (the Sun), and that the Sun, transmitter and receiver are in the $z=0$ plane. $R$ is the radius of the Sun, $x_R$ and $r_R$ are the receiver $x$-coordinate and distance from the origin, $x_T$ and $y_T$ are the transmitter $x$- and $y$-coordinates, and $r_T$ is the transmitter distance from the origin (see figure \[tofeqdiagram\]).

![\[tofeqdiagram\] Diagram of coordinate system used for equation \[tofeq\].](tofeqdiagram.eps){width="6in"}

The sizes of each term in eq. \[tofeq\] for photons travelling 2 AU and just grazing the limb of the Sun are given in table \[sttab\], also taken from @1983PhRvD..28.3007R.

                                      Term:                                       Size of term in seconds:
  ------------------------------------------------------------------------------ --------------------------
                                  $4\frac{r}{R}$                                           $2000$
                 $2(1+\gamma)\frac{GM}{R}\ln (4(\frac{r}{R})^2)$                    $2.4 \times 10^{-4}$
                                 $4\frac{GM}{R}$                                    $2.0\times 10^{-5}$
                   $4(1+\gamma)^2(\frac{GM}{R})^2 \frac{r}{R}$                      $3.6\times 10^{-8}$
                         $ 2(1+\gamma)J_2 \frac{GM}{R} $                          $ 4.9 \times 10^{-10} $
   $4[2(1+\gamma)-\beta+\frac{3}{4}\delta](\frac{GM}{R})^2\arctan(\frac{r}{R})$   $ 2.5 \times 10^{-10} $
         $2(1+\gamma)(\frac{GM}{R})^2 \frac{R}{r} \ln (4(\frac{r}{R})^2)$          $2.4 \times 10^{-12} $
                    $2(2\beta-1)(\frac{GM}{R})^2 \frac{R}{r}$                       $9.7\times 10^{-14}$

  : \[sttab\] Size of terms in time-of-flight expression for 2 AU light path which grazes the limb of the Sun. PPN parameters are set to their values in GR, and $J_2$ and $J_z$ are set to their values in Dicke’s model of the Sun.

The sizes of these terms should be compared with the timing accuracy expected for the LATOR interferometer measurement, remembering that the actual differential times-of-flight measurements for the two LATOR signals are a fraction of the displayed terms in table \[sttab\].

The leading term, after the 2000-second geometrical time-of-flight term, is the term proportional to $1+\gamma$, of size $2.4\times10^{-4}$, with its second-order correction of size $3.6\times10^{-8}$. This is the effect whose measurement will give PPN parameter $\gamma$ from the LATOR measurements. The next largest term, with size $4.9\times10^{-10}$ enables the solar quadrupole moment, $J_2$, to be measured, while the term of size $2.5\times10^{-10}$ will give a linear combination of $\gamma$, $\beta$, and the post-post-Newtonian term $\delta$.

The fundamental current incompatibility of General Relativity and Quantum Mechanics suggests that the PPN parameters should deviate from their values in GR at some level. The size of the deviation of the PPN parameters from their value in GR is theory dependent. @1993PhRvL..70.2217D [@1993PhRvD..48.3436D] suggest that for some scalar-tensor theories of gravity the deviation in the PPN parameter $\gamma$ occurs at the level $1-\gamma \sim
10^{-7}$, significantly smaller than the uncertainty in the current best estimate for $\gamma$, at $\sim 10^{-5}$ (see @Bertotti:2003rm). The primary goal of the LATOR mission is measurement of the PPN parameter $\gamma$ to an accuracy of $10^{-9}$. Secondarily, LATOR will provide the first measurement of a post-post-Newtonian parameter, $\delta$, and a measurement of the solar quadrupole moment parameter $J_2$, which will provide information about the solar interior.

Covariance Study
================

We model the LATOR instrument in the solar system and use a linear least squares covariance analysis to solve simultaneously for the uncertainties in the PPN parameters and orbit parameters. A Newtonian model is used for the transmitter orbits, and time delays for the interferometer legs are modeled using eq. \[tofeq\]. [*A priori*]{} uncertainties for the orbit parameters are assumed, consistent with typical uncertainties in spacecraft radio tracking. The transmitter coordinates are evaluated as functions of the time and of the orbit parameters for each spacecraft, while receiver coordinates are evaluated as functions of time using a simple circular heliocentric orbit. A justification for this simple model of the transmitter and receiver orbits is given in appendix \[orbjust\]. The interferometer is assumed to be inertially oriented. We model two constant body-fixed accelerations for each transmitter, one along the Earth line-of-sight and one perpendicular to it. For each assumed data point, the time-of-flight from each spacecraft to each antenna is calculated as a function of the time of observation and of the parameters of the model. The parameters include the constants of motion for the transmitter orbit (one set for each transmitter), the DC accelerations on each spacecraft (one set per transmitter) and the PPN parameters. These parameters are listed in table \[prms\]. The goal of the covariance analysis is to estimate the uncertainties with which these parameters will be determined in the final LATOR mission data analysis.

                Parameter               Description
  ------------------------------------- ------------------------------------------------------------------------------------------------------------------------------
              $a_1$, $a_2$              Semi-major axes for orbits of transmitting spacecraft 1 and 2.
              $e_1$, $e_2$              Eccentricities for orbits of transmitting spacecraft 1 and 2.
        $\phi_{01}$, $\phi_{02}$        Initial phase for orbits of transmitting spacecraft.
         $\omega_1$, $\omega_2$         Angle of semi-major axes of transmitter orbits with respect to x axis.
       $a_{\perp 1}$,$a_{\perp 2}$      Constant acceleration in drag-free system, perpendicular to transmitter-receiver line-of-sight, for transmitting spacecraft.
   $a_{\parallel 1}$,$a_{\parallel 2}$  Constant acceleration in drag-free system, parallel to transmitter-receiver line-of-sight, for transmitting spacecraft.
                $\gamma$                Space-time curvature per unit rest mass.
                $\delta$                Time curvature per unit gravitational self energy.
                  $J_2$                 Solar quadrupole moment parameter.

  : \[prms\] Parameters to be solved for in covariance study

Notably absent from the list of parameters is the parameter $\beta$. This is because the only significant term in the time-of-flight equation which contains $\beta$ (see table \[sttab\]) is $$4[2(1+\gamma)-\beta+\frac{3}{4}\delta](\frac{GM}{R})^2\arctan(\frac{r}{R})$$ in which $\beta$, $\delta$ and $\gamma$ are all completely correlated. Other terms give $\gamma$, but the only other term to contain $\beta$ is smaller, meaning that $\beta$ would be less well determined than the combination $\beta+3/4\delta$ and that the incertainty in both would be that of $\beta$ alone. However, since $\beta$ is already known from other solar system experiments (@2004PhRvL..93z1101W) with an accuracy (see table \[ppnsig\]) better than the best LATOR can manage for this term, our analysis assumes that $\beta$ is already known. The term $$\nonumber
4[2(1+\gamma)-\beta+\frac{3}{4}\delta](\frac{GM}{R})^2\arctan(\frac{r}{R})$$ is therefore used to solve for $\delta$ alone.

At each time sampled, the LATOR signal is formed from $$S = [\tau_{12}-\tau_{11}]-[\tau_{22}-\tau_{21}]\label{signaleq}$$ where each $\tau$ is the time-of-flight for a particular transmitter-receiver pair; so, for example, $\tau_{12}$ would be the time-of-flight between transmitter 1 and receiver 2. At each time, the contribution from the current time step to the information matrix is formed by calculating the derivatives of $S$ with respect to the parameters of table \[prms\]. Referring to the parameter set collectively as $\vec{x}$, the contribution to the information matrix at each time $t_i$ is given by $$\alpha_{jk}(t_i) = \frac{1}{(\sigma)^2}\frac{\partial S(t_i,\vec{x})}{\partial x_j} \frac{\partial S(t_i,\vec{x})}{\partial x_k}\label{fmcon}$$ where $\sigma$ is the uncertainty in LATOR’s measurement of each time delay. In our analysis, $\sigma$ was set to $3\times10^{-20}$ s, corresponding to the conservative 10 picometer interferometer uncertainty estimate found in LATOR literature (see, for instance, @0264-9381-21-12-001).

The derivatives in eq. \[fmcon\] were calculated using analytic derivatives for PPN parameters, finite differencing for orbit constants of motion, and finite differencing with a perturbative Enke integration for the DC accelerations. Calculations leading up to the time-of-flight calculation were performed in double precision. Round-off error was a serious problem for derivatives with respect to orbit parameters, so we used Ridders’ extrapolative finite differencing algorithm (from @nr chapter 5) and performed all calculations after the time-of-flight calculation in the GNU Multiple Precision Arithmetic Library (GMP) in order to keep sufficient precision to solve for uncertainties in all of the variables.

The terms in the information matrix, $\alpha_{jk}(t_i)$, were then summed over a period of about 1 year, during which the transmitting spacecraft pass behind the Sun three times as viewed from Earth (see figure \[orbitplot\]). We used a 1-minute sampling cadence, but the number of data points was further reduced because the transmitting spacecraft are only visible from Earth for half of the ISS hour-and-a-half orbit, and it may take tens of minutes per orbit for the receiving interferometer to acquire lock. This was simulated by adding to the information matrix for only 10 minutes out of a 90 minute period. The resulting integration has about 50000 time steps. The total information matrix is $$\alpha_{jk} = \sum_i \frac{1}{(\sigma)^2}\frac{\partial S(t_i,\vec{x})}{\partial x_j} \frac{\partial S(t_i,\vec{x})}{\partial x_k}\label{fm}$$

The information matrix was inverted using a Singular Value Decomposition (SVD), and the diagonal elements of the inverse gave the standard variations for the corresponding parameters: $$\sigma_i=\sqrt{\alpha^{-1}_{ii}} \qquad \qquad \textrm{(no sum)}\label{vareq}$$

The Impact Parameter Problem {#ipsec}
============================

Since the leading $\gamma$ term in the time-of-flight formula is logarithmic, its uncertainty scales as $1/r$. We will so far anticipate the results of our analysis to say that we expect $\gamma$ to be determined to an accuracy $3 \times 10^{-9}$. If this is to be accomplished, then the size of the coefficient of $\gamma$ must be known correctly to the same accuracy. For signals passing close to the Sun, the $1+\gamma$ term in eq. \[tofeq\] may be approximated as $$\label{approx}
(1+\gamma)\frac{GM}{R}(1-\frac{GM}{R}\frac{R}{r_T})\ln\frac{x_R+r_R}{x_T+r_T}
\approx (1+\gamma)\frac{GM}{R}\ln(4r_Rr_T/d^2)$$ where $d$ is the impact parameter. Thus, if the coefficient of $\gamma$ is to be known to $3 \times 10^{9}$, the impact parameter must be known to a part in $10^9$. For paths passing near the solar limb, the impact parameter is $\sim10^{9}$ m, meaning that it must be known to within a meter. How is this to be done?

First, we point out that the spacecraft-to-spacecraft link that could be used to give the base of the narrow triangle linking the two spacecraft and the interferometer provides no information on how that triangle is oriented relative to the Sun. Observations of the solar limb using the interferometer cannot possibly be made accurate to 1 meter, and, even then, the distance from the solar limb to the center of the Sun (for purposes of calculating the solar potential) cannot possibly be modeled. One could conceive of doing orbit determination using some typical Doppler or ranging tracking of the spacecraft or even of using the precise relative angular measurements from the interferometer themselves as a new navigation data type. However, experience with orbit determination for free-flying interplanetary spacecraft shows that non-gravitational forces do not allow anything like 1-meter accuracy over periods of months. If the impact parameter problem is to be solved, then non-gravitational forces must be eliminated using a drag-free system on board each spacecraft.

In our covariance study, we have assumed that the LATOR spacecraft follow purely gravitational orbits (except for a constant body-fixed acceleration that is solved for in the covariance study). We will find in our study that not only is the impact parameter problem solved by the subsequent orbit determination, but that the spacecraft-to-spacecraft distance problem is likewise resolved. This is to say that, as we simultaneously solve for orbit parameters for both spacecraft along with the PPN parameters, any uncertainties in the inertial positions of the spacecraft are correctly taken into account through the correlations in the covariance matrix. Thus, whatever final uncertainty we derive for the PPN parameters will be the correct uncertainty, given the uncertainties in the data-derived orbits of the spacecraft.

To determine how good a drag-free system needs to be to enable a pure gravitational trajectory to be a good approximation, let us consider the following:

1.  A $10^{-13}$-radian angular accuracy at a distance of $3\times10^{11}$ m corresponds to a position accuracy of 3 cm.

2.  The position noise for a spacecraft suffering acceleration noise with noise spectral density $S_a$ is $x_{\rm
    rms}=\sqrt{S_a/(16\pi^4f^4)}$, at a Fourier frequency $f$.

3.  Therefore, if we require 3 cm accuracy at a frequency 1/month ($4\times10^{-7}$ Hz), we derive a drag-free requirement of $S_a=3.6\times10^{-26}$ m$^2$s$^{-4}$Hz$^{-1}$, or a root spectral density $\sqrt{S_a}=1.9\times10^{-13}$ ms$^{-2}$Hz$^{-1/2}$.

The currently planned LISA reference sensors are expected to have an accuracy of $3\times 10^{-15}\mathrm{m}\mathrm{s}^{-2}\mathrm{Hz}^{-1/2}$. Below $10^{-4} \mathrm{Hz}$, a $1/f$ increase is expected due to thermal drift resulting from the passive thermal isolation of the reference sensors. At $3\times 10^{-8} \mathrm{Hz}$, this would produce acceleration noise of $10^{-11} \mathrm{m}\mathrm{s}^{-2}\mathrm{Hz}^{-1/2}$, so a mK active thermal control (not a difficult technology) would be needed to keep thermal noise sources below the $2\times 10^{-13}\mathrm{m}\mathrm{s}^{-2}\mathrm{Hz}^{-1/2}$ we have assumed for the LATOR accelerometer. Voltage drifts in the measurement electronics do not affect the capacitance bridge used to read the accelerometer, since it is nearly a null measurement. Proof-mass charge control is absolute and independent of long-period effects. Thermal drift of the non-controlled parts of the spacecraft, along with mechanical relaxation of the spacecraft structure, [*will*]{} produce long-term self-gravity changes, but we have calculated these to be well out of the way of the level of performance we have assumed, as long as there is a level of spacecraft mechanical and thermal engineering comparable to that already required for LISA. Most other noise sources are white and LISA performance would keep them below the $10^{-15}\mathrm{m}\mathrm{s}^{-2}\mathrm{Hz}^{-1/2}$ LISA requirement.

Results and Conclusions
=======================

The predicted accuracies of orbit parameters and spacecraft DC accelerations from one year of LATOR data are shown in table \[nuisres\]. The 60-cm uncertainty in the two semi-major axes is primarily the result of strong correlation between these parameters. (The uncertainty for $a_1$ or $a_2$ would have been about 60 microns if one of these two parameters were being solved for alone.)

                Parameter                         LATOR predicted uncertainty
  ------------------------------------- -----------------------------------------------
              $a_1$, $a_2$                                   1.4 m
              $e_1$, $e_2$                            $2 \times 10^{-11}$
        $\phi_{01}$, $\phi_{02}$                     $1.3 \times 10^{-10}$
         $\omega_1$, $\omega_2$                       $4 \times 10^{-11}$
       $a_{\perp 1}$,$a_{\perp 2}$       $4 \times 10^{-14}$ $\textrm{m}/\textrm{s}^2$
   $a_{\parallel 1}$,$a_{\parallel 2}$   $3 \times 10^{-14}$ $\textrm{m}/\textrm{s}^2$

  : \[nuisres\] Covariance study estimates for LATOR determination of uncertainties in transmitter constants of the motion and DC accelerations

   Parameter  Significance                                         Current best estimate                                                       LATOR predicted uncertainty
  ----------- ---------------------------------------------------- --------------------------------------------------------------------------- -----------------------------
   $\gamma$   Space-time curvature per unit rest mass.             $ 1 + (2.1 \pm 2.3)\times 10^{-5} $[@Bertotti:2003rm]. Equal to one in GR   $2.7\times 10^{-9}$
    $\beta$   Space curvature per unit gravitational self-energy   $1+(0.9\pm 1.1)\times 10^{-4}$[@2004PhRvL..93z1101W]. Equal to one in GR    Not solved for.
   $\delta$   Time curvature per unit gravitational self-energy    Unmeasured. Equal to one in GR                                              $1\times 10^{-3}$
     $J_2$    Solar quadrupole moment parameter.                   Umeasured. Estimate of about $10^{-7}$ inferred from solar models.          $5\times 10^{-9}$

  : \[ppnres\] Predicted uncertainties in PPN parameters with LATOR and comparison with current best estimates.

The results for the PPN parameters are shown in table \[ppnres\]. The uncertainty in $\gamma$ will represent a four order-of-magnitude improvement over the current Cassini results and will, for the first time, approach the level where heuristic string-theory arguments predict that a deviation from GR might be expected. The Parametrized Post-Post-Newtonian (PPPN) parameter $\delta$ will be measured for the first time. This is important because there are several theories whose PPN parameters are identical with GR, but which differ from GR in their predictions for the PPPN $\delta$ term. Finally, we predict a 6% determination of the solar quadrupole moment, relative to its expected value of $10^{-7}$. This is a direct, model-independent value and may be used as a touchstone value for various models of the solar interior.

We conclude that LATOR will represent a significant step forward in the experimental tests of relativistic theories of gravity and will be able to satisfy the goals that were suggested in the initial project proposals. We find that, with a $1.9\times10^{-13}$ ms$^{-2}$Hz$^{-1/2}$ drag-free system covering a band down to nearly DC, it will be possible to simultaneously extract the PPN parameters and all relevant orbit parameters from the interferometer data alone. This seems to be the only good solution to the impact parameter problem, and we strongly suggest that the engineering studies for LATOR consider this hardware option.

Justification of Orbit Modelling {#orbjust}
================================

The assumed accuracy of each interferometer measurement is $10^{-13}$ radians, which, at a distance of 2AU ($3\times 10^{11}$m) corresponds to 3 cm along-track position accuracy. As a comparison, let us consider the recent range data to various Mars spacecraft, the forces on Mars being purely gravitational. The data for the last decade have post-fit residuals of 1-2 m. The parameters that enter the solution include spacecraft locations and MarsÕ physical ephemeris for landers, orbital elements and MarsÕ gravity field for orbiters, corrections to Earth tracking station locations, solar corona and troposphere delays, orbits and masses of the planets, and the masses of several asteroids. This combination of $\sim 200$ parameters in comparison with our 4 orbital elements would seem to suggest that we have oversimplified the problem. However, the LATOR data and model are immune to the effects of most of these elements of the model.

The LATOR interferometers are in orbit above the troposphere. The laser frequencies are high enough to avoid significant plasma propagation delays. There is no local disturbing planet to complicate the spacecraft motion. The interferometer makes an [*angular*]{} position measurement, so errors in the interferometer position cancel to first order. Errors in the interferometer orientation likewise cancel when the differential position of the two spacecraft is calculated. Finally, the long-period solar system model uncertainties in the position of each spacecraft are significantly reduced when the [*differenced*]{} spacecraft positions are calculated. For an along-track sinusoidal error for each spacecraft of size $s$ and period $T$ (in days), the size of the error in the differenced position is given by $$\label{dperror}
\Delta s = \frac{s \cos(2\pi t/T)}{vT}\delta \lambda$$ where $v$ is the velocity of the spacecraft in degrees per day and $\delta\lambda$ is the difference in heliocentric longitude between the two spacecraft, in degrees. Since $v\sim 1$ and $\delta\lambda\sim 2^{\circ}$, and since the solar system model errors are of order $\sim$years, the error in the differential position of the two spacecraft will be reduced by a factor $\frac{\delta \lambda}{T}$ (which is $\sim 0.01$ or less) relative to the single-spacecraft position error. Thus a spacecraft error of 1-2 m becomes an error of 1-2 cm in the LATOR observable, slightly below the 3 cm error we assume in our covariance study. It is therefore not an unreasonable assumption to treat the solar system parameters as known to this level, by virtue of the present and ongoing accurate Mars ranging data.

We would like to thank Slava Turyshev, Mike Shao and Ken Nordtvedt for help understanding the LATOR design. We would also like to thank Slava Turyshev for reviewing our manuscript in advance of submission.
