THE ORBITS AND MASSES OF THE MARTIAN SATELLITES AND THE LIBRATION OF PHOBOS

The model for the satellite orbits is a numerical integration of their equations of motion (Peters 1981). The formulation is in Cartesian coordinates centered at the Martian system barycenter and referenced to the International Celestial Reference Frame (ICRF). The modeled dynamics include the asphericity of Mars, the mutual interactions of Phobos and Deimos, the perturbations due to the Earth, Moon, Jupiter, Saturn, and the Sun, the Phobos figure, and the tide raised on Mars by Phobos. We ignore the tide raised by Deimos; because of Deimos's smaller mass and larger distance from Mars, the tidal acceleration caused by Deimos is roughly 3 orders of magnitude smaller than that caused by Phobos. Moreover, to date there has been no conclusive observational evidence of a tidal effect on Deimos. We do, however, account for the effect of the tide due to Phobos on the motion of Deimos (our software applies active tidal forces to all bodies being integrated). In modeling the acceleration induced by the Phobos figure, we assume that Phobos is in synchronous rotation with its pole normal to its orbit plane and its prime meridian librating about the sub-Mars direction (i.e., a libration in longitude); the libration in latitude is ignored.

JPL planetary ephemeris DE421 (Folkner et al. 2008) provides the positions and GMs of the Sun, Moon, and planets. The Love number, gravity field, and orientation of Mars are from Konopliv et al. (2006). We truncated the gravity field at degree 8 in the zonal harmonics and fifth degree and order in the tesseral harmonics; tests showed higher degree and order had only meter level effects on the integration. Because of software limitations, we could not use the Mars orientation model as published. Instead, we fit a Fourier series to the ICRF pole right ascension and declination and the prime meridian derived from the model. The series, given in the Appendix, reproduces the angles at the 6 mas level which is well below their actual uncertainty. The Phobos quadrupole field gravitational harmonics (J₂ and C₂₂) in the figure acceleration are from Borderies & Yoder (1990).

In order to process the Viking and Phobos 2 tracking data, we also had to model the spacecraft orbits. These orbits are numerically integrated with JPL's Orbit Determination Program (ODP; Moyer 1971; Ekelund 1979), which is the software set used at JPL for planetary spacecraft navigation. Like the satellite orbits, the spacecraft orbits are represented by Cartesian coordinates centered at the Martian system barycenter and referenced to the ICRF. The gravitational model for the spacecraft is the same as that for the satellites except we did not truncate the Mars gravity field or approximate the Mars orientation (the published gravity field and orientation model are designed to be used in the ODP). We ignored the effect of the Phobos tide on the spacecraft but included the Phobos figure acceleration, i.e., Phobos was not treated as a point mass during the spacecraft flybys. Because the flybys are close to Phobos, the Phobos figure causes a small disturbance in the spacecraft orbits; the figure effect is small enough that could have been neglected, but we retained it for dynamical completeness.

All three spacecraft were affected by the force due to solar radiation pressure. We accounted for this force with a simple bus model characterized by effective areas along and normal to the spacecraft–Sun direction. The spacecraft were three-axis stabilized, i.e., their orientation was changed and maintained by attitude control thrusters. Consequently, their motion was subject to small accelerations due to thruster imbalances and control system leakage; we included those effects in our force model as small accelerations along the spacecraft body axes. We have no detailed history of the actual attitudes of the spacecraft body axes. However, we know that the Viking axis of symmetry was normally directed toward the Sun, and the other two axes were set with the aid of a reference star. Phobos 2 was also oriented relative to the Sun and stars. For modeling purposes, we adopted the spacecraft–Sun direction as one body axis for all three spacecraft. The direction to the star Canopus defined the roll angle about that axis.

On 1989 March 25, Phobos 2 was reoriented in order to take pictures of Phobos (the camera was fixed to the spacecraft which necessitated turning it to point the camera). The disturbances induced by the turns were modeled as an impulsive maneuver on that date.

We fit the orbits, in a weighted least-squares sense, to the observational data by adjusting the values of the parameters in our dynamical and observational models. The dynamical parameters are: (1) the epoch position, velocity, and GM of each satellite, (2) the amplitude of the Phobos longitude libration (see Section 4.3), (3) the Phobos tidal bulge lag angle (see Section 4.4), (4) the epoch states, solar pressure coefficients, and non-gravitational accelerations for the Viking and Phobos 2 spacecraft, and (5) the Phobos 2 impulsive maneuver.

The observational model parameters are as follows.

• 1.  
  The scale and orientation of the CCD used for the astrometry of Colas (1992), Colas & Arlot (1991) discuss the CCD calibration.
• 2.  
  Adjustments to the Mariner 9, Viking, and Phobos 2 trajectories associated with the respective imaging data; for the Mariner 9 imaging acquired on the approach to Mars, we corrected the direction of the spacecraft's trajectory (i.e., the inertial direction of the asymptote of the hyperbolic conic approximation to the trajectory); for the Mariner 9, Viking, and Phobos 2 imaging acquired in orbit about Mars, we corrected the spacecraft orbit node on the plane normal to the Earth–Mars line of sight; for the Viking and Phobos 2 imaging, we also corrected the time of spacecraft orbit periapsis passage. See Duxbury & Callahan (1988, 1989) and Kolyuka et al. (1991) for details on the spacecraft orbit adjustment procedures.
• 3.  
  The camera pointing angles and satellite image phase biases for the MRO data, the known locations of background stars in the pictures specify the camera pointing, the image phase biases account for lighting effects which introduce errors in the determination of the center of figure of the extended satellite images. The phase biases are directed toward the Sun with magnitude a b_jsin^j(ϕ/2) where a is the radius of the satellite, b_j is the estimated phase bias scale factor, and ϕ is the solar phase angle. We included the three bias terms for j = 0, 1, 2.
• 4.  
  Corrections to the spacecraft tracking data ionosphere calibrations to account for inaccuracies in the delays computed from Bent model discussed in the previous section.
• 5.  
  Corrections to the spacecraft tracking data troposphere calibrations to account for the daily variations in troposphere delay.
• 6.  
  Biases in the Viking Doppler data; previous work with the Viking data (Smith et al. 1993; Yuan et al. 2001) found that they contained systematic errors which could be mitigated by including estimated biases.
• 7.  
  The tracking station locations associated with the Phobos 2 data; the station locations contained small systematic errors because the data used to determine them were not completely calibrated (Kroger et al. 1992).

We were unable to determine the Phobos J₂ and C₂₂ as part of our orbit fit and conclude that there is insufficient information in the data set to isolate these parameters.

We grouped the Earth-based astrometric data according to type, observatory, and the observing period in which they were acquired. Data weights for each group were assigned through an iterative process to be consistent with the root mean square (rms) of the residuals (the differences between the actual observations and their values predicted by our model) of that group. Analogous to the Earth-based astrometry, the MGS, MRO, and MEX spacecraft imaging data and MOLA transit data were given weights consistent with the rms of their respective residuals.

For the imaging observations from Viking and Phobos 2, we initially tried the weights suggested in the publications. However, after looking at the residuals we concluded that the weights were too pessimistic. Ultimately, we tightened the Viking 1 weights by 75%, the Viking 2 weights by 65%, and those for Phobos 2 by 85%.

We based the Viking Doppler data weights for each tracking pass on the rms of the residuals for that pass scaled by a factor to account for the fact that the Doppler noise is not a white-noise process (Folkner 1994). The scale factor is 0.468(86400/τ)^1/3 where τ is the Doppler sample interval in seconds (for a 10 minute sample interval the factor is 2.45). This scaling preferentially weights the Doppler at the diurnal frequency (86,400 s); however, it yields a weight which is too conservative for the satellite close encounter data. For the Phobos and Deimos encounter passes, we employed the weighting algorithm (Mackenzie & Folkner 2006) which was developed for Cassini planetary gravity analysis; the same algorithm was applied to all of the short duration Phobos 2 tracking passes as well. The algorithm accounts for the noise structure of the tracking data, but it is designed for short arcs of tracking data where the characteristic signature in the data is that of the gravity parameters in a close planet or satellite flyby.

To fit the weighted observations, we used a batch-sequential square-root information filter with a number of parameters treated as white-noise stochastic parameters, i.e., parameters which are constant within a batch but vary from batch to batch. See Bierman (1977) for a discussion of square-root information filtering. Our stochastic parameters were: (1) Colas's CCD scale and orientation batched according to observation series (Colas & Arlot 1991); (2) the Mariner 9, Viking, and Phobos 2 orbit node batched by orbit about Mars; (3) the Viking and Phobos 2 periapsis timing batched by orbit about Mars; (4) the MRO camera pointing angle batched by picture; (5) the spacecraft tracking data ionosphere and troposphere calibration corrections batched by day; (6) the Viking Doppler biases batched by tracking pass; and (7) the spacecraft non-gravitational accelerations batched by the times of Mars periapsis and apoapsis for Viking 1 and Phobos 2 and the times of apoapsis for Viking 2.

Table 1 gives the postfit residual statistics for the observations grouped by source (although the original source is provided, most of the pre-1982 observations appear in Morley's catalog). For those observations which were made in the form of position angles, we scaled the residuals by the separation distance before forming the rms. The residuals for spacecraft based data were converted to units of km depending upon the spacecraft satellite range at time of the measurement. For the purpose of comparison, the Earth-based astrometry can be converted to km with the factor 375 km arcsec⁻¹. Examination of the statistics shows that the best Earth-based astrometry (Veiga 2008) fits to the 19 km level and the best spacecraft imaging (Willner et al. 2008) fits to the 0.5 km level. Although the statistics cannot be compared directly to those in Table 3 in Jacobson et al. (1989) which groups the rms by opposition rather than source, we have verified that for the pre-1988 data our new orbits provide a fit comparable to our previous orbits.

Table 1. Observation Residual Statistics

Source	Number			rms
	Pho.	Dei.	D-P	Pho.	Dei.	D-P
Hall (1878)	79	108		0782	0742
Pickering (1882)	110	290		0753	0825
Pritchett (1878)	11	11		0713	1031
Common (1878)		23			1069
Christie et al. (1878)		6			0839
Hall (1880)	81	90		0337	0521
Pritchett (1880)	10	5		1113	1181
Common (1879)	43	60		0875	0880
Lindsay (1879)		19			0855
Pritchard (1880)		10			0566
Maunder (1880)		8			0862
Hall (1882)	13	21		0523	0584
Common (1882)	9	10		0577	0908
Hall (1884)	4	19		0819	0531
Brown (1891)		10			0621
Keeler (1888)	115	52	1	0398	0610	0430
Hall (1888)	4	21		0483	0561
Keeler (1890)	47	26		0310	0590
Hall (1891)	9	12		0684	0975
Campbell (1892)	93	87		0405	0517
Hall (1893)	35	37		0714	0956
Campbell (1895)	398	345		0253	0299
Newall (1895)	77			0644
Barnard (1897)	113	30		0423	0479
Schaeberle (1897)	51	65		0430	0618
Kostinsky (1897)		4			0222
Hussey (1898)	60	20		0276	1114
Struve (1898)	79	76	1	0226	0506	0200
Wirtz (1909)	4	10		0668	0720
USNO (1911)	97	137		0384	0513
Aitken (1909)	48	40		0308	0330
Barnard (1910)	14	13		0304	0214
Wirtz (1912)	8	10	6	0448	0759	0393
Kostinsky (1913)	48	40		0591	0739
USNO (1929)	416	377		0407	0431
Jeffers (1925)	101	70		0362	0388
USNO (1954)	50	26	43	0432	0536	0242
Rizvanov & Shshukin (1969)		22			0770
Karimova & Kholopov (1969)		4			0588
Kovalenko (1969)		7			1035
Fatchikhin (1970)		63			0769
Deutsch (1970)	8	52		0489	0691
Bashtova & Rakhimov (1973)		21			0985
Khatisov & Salukvadze (1974)	39	44		0731	0721
Khatisov & Salukvadze (1975)	118	228		0542	0569
Kovalenko et al. (1980)	26	90		0848	0690
van Biesbroeck (1970)			44			0325
Kiseleva (1976)			34			0437
Tolbin (1985)	12	26		0850	0614
A. T. Sinclair (1988, private communication)			22			0533
Kiseleva & Chanturiia (1988)	12			0609
Duxbury & Callahan (1988)	326	212		8.54 km	4.00 km
D. Pascu (1988, private communication)	302	420	112	0156	0165	0200
Duxbury & Callahan (1989)	102	62		10.85 km	10.84 km
Kiselev et al. (1989)	224		100	0908		0745
Jones et al. (1989)	172	24	126	0134	0137	0183
Shor (1989)	280	160	213	1111	1020	0567
Kalinichenko et al. (1990)	28	2	28	0321	0096	0119
Tel'nyuk-Adamchuk et al. (1990)	190	88	56	0942	1084	0774
Bobylev et al. (1991)	324	104	276	0486	0668	0418
Nikonov et al. (1991)	46			0530
Kolyuka et al. (1991)	74	14		0.65 km	9.93 km
Colas (1992)			1626			0243
Kudryavtsev et al. (1992)			1198			0118
C. E. Hildebrand (1998, private communication)	4			3.06 km
V. Golovnya (2001, private communication)	72	349	188	0879	0779	0473
W. M. Owen (2003, private communication)		6	12		0224	0169
D. Pascu (2004, private communication)	14	10	414	0379	0219	0062
Bills et al. (2005)	32			2.72 km
Oberst et al. (2006)	52	20		1.64 km	4.90 km
S. P. Synnott (2006, private communication)	103	478		6.96 km	5.33 km
Veiga (2008)	282	304		0056	0051
Willner et al. (2008)	138			0.49 km

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Figures 1–3 display the residuals for the respective Viking 1, Viking 2, and Phobos 2 Doppler tracking. No obvious signatures remain, and the scatter is reasonable for S-band and C-band tracking. The apparent noise increase at the Deimos encounter (1977 October 15, 11:15:00) is a consequence of the reduction of the Doppler sample interval from 1 minute to 10 s.

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Table 2 contains the epoch state vectors determined from the fit; the vector components are specified to sixteen digits to facilitate future orbit integrations.

Table 3 summarizes a number of previous determinations of the satellite GMs together with our results.

The first estimates of the Phobos GM using Viking 1 tracking data were by Christensen et al. (1977), who deduced the GM from the changes in the mean orbital period of the spacecraft caused by 14 consecutive close Phobos flybys, and by Tolson et al. (1977), who fit the spacecraft orbit and the GM directly to the data from the three closest spacecraft flybys. Hildebrand et al. (1979) made the initial Deimos GM estimate with 1 hr of tracking acquired during the Viking 2 flyby. Williams et al. (1988) later obtained both GMs using the combined tracking from 8 Viking 1 Phobos flybys and the Viking 2 Deimos flyby. Konopliv et al. (2006) found the GMs as part of his determination of the Martian gravity field by including data from 29 Viking 1 Phobos flybys and the Viking 2 Deimos flyby.

Berthias (1990) estimated the Phobos GM in a fit to the Phobos 2 tracking data over the period 1989 February 18 to 1989 March 27; he also estimated the spacecraft orbit, maneuvers, and the Phobos ephemeris. Kolyuka et al. (1990) carried out a similar fit but restricted his data span to the period of the QSO. Shishov (2008) processed the tracking data within the context of his development of a numerically integrated Phobos ephemeris.

Smith et al. (1995) took a different approach to the use of spacecraft tracking; they inferred the satellite GMs from the satellites' contribution to the long period secular perturbations of the Mariner 9 and Viking spacecraft orbits.

Rosenblatt et al. (2008a) applied the same approach as Smith et al. (1995) but to the MEX spacecraft which, because of its orbital geometry, was more sensitive to the satellite perturbations. Rosenblatt et al. (2008b) later extended that analysis by adding the tracking data from a close Phobos flyby by MEX. Andert et al. (2008) obtained their result with the MEX tracking from two close flybys of Phobos.

Our Deimos GM is in agreement with all previous results. Its uncertainty is smaller than those associated with the earlier determinations from the Viking 2 tracking, most likely a consequence of our data weighting strategy. The values found from the MEX data are a good match for those based on the Viking 2 data but are considerably less well determined because MEX has had no close Deimos flybys.

There is considerable variation among the various Phobos GMs. This is not too surprising because, unlike with Deimos, data from multiple satellite flybys contribute to the Phobos GM determination. Consequently, dynamical modeling of both the spacecraft and satellite motions have a significant effect on the result. All of the pre-2006 work employed Mars gravity fields derived from Mariner 9 and Viking data (Gapcynski et al. 1977; Christensen & Balimino 1979). The analysis by Shishov (2008) used the field of Lemoine et al. (2001) based on MGS data, and the remaining post-2006 work relied on the field of Konopliv et al. (2006). Prior to 2008, the orbit of Phobos was represented by analytical theories (Sinclair 1972; Born & Duxbury 1975; Ivanov et al. 1988; Morley 1990; Chapront-Touzé 1990). Tolson et al. (1977), however, fit a numerical integration to the Born–Duxbury theory and used the integrated orbit when processing the spacecraft tracking. For the analysis of the MEX tracking, Rosenblatt et al. (2008a, 2008b) took the integrated ephemeris of Lainey et al. (2007) whereas Andert et al. (2008) chose the integrated ephemeris of Jacobson (2008). As stated earlier, Shishov (2008) has developed his own integrated orbit.

All previous investigators allowed for solar radiation pressure on the various spacecraft. However, in work involving either the Phobos 2 or Viking data, only Kolyuka et al. (1990) and Konopliv et al. (2006) considered the effects of unmodeled non-gravitational accelerations. The former found the accelerations on Phobos 2 to be negligible, and the latter added small impulsive maneuvers to his Viking data processing to account for unmodeled forces. Table 3 contains our GM values based on only the Viking tracking and only the Phobos 2 tracking for cases in which those accelerations were either excluded or included. It is clear that the accelerations have little effect on Phobos 2 but have a major effect on Viking 1. Ignoring the Viking 1 accelerations yields a Phobos GM closer to the value found with the Phobos 2 data alone and drastically reduces the uncertainty. Accounting for the accelerations leads to an increase in both the value and uncertainty; however, a statistical agreement remains between the two single spacecraft determinations. The GM found with the combined data set, our adopted value, is essentially the same as that found with the Phobos 2 data because the strength of the Viking 1 data is significantly weakened by the non-gravitational accelerations.

Under our assumptions that Phobos is in synchronous rotation with its pole normal to its orbit plane and its axis of minimum principal moment of inertia directed toward Mars, the potential for the acceleration of Mars due to the Phobos quadrupole gravity field is:

$$\begin{equation} U = \frac{1}{2}\,\mu _{1}\left(\frac{R^{2}}{\rho ^{3}}\right) (J_{2}+6\,C_{22}\,\cos 2\theta), \end{equation} \tag{ 1 }$$

where

μ₁ = GM of Phobos,

R = equatorial radius of Phobos,

ρ = distance from Phobos to Mars,

J₂ = second degree zonal harmonic of Phobos,

C₂₂ = second degree and order tesseral harmonic of Phobos, and

θ = the angle between the direction from Phobos to Mars and the axis of minimum principal moment of inertia.

Because Phobos's rotation rate matches the orbit's mean longitude rate (synchronous rotation), and because Phobos is assumed to librate about the sub-Mars direction:

$$\begin{equation} \theta = f - M + \tau, \end{equation} \tag{ 2 }$$

where f is the true anomaly, M is the mean anomaly, and τ is the forced libration angle. The Phobos-forced libration takes the form (Duxbury 1977):

$$\begin{equation} \tau = \mathcal {A}\sin M. \end{equation} \tag{ 3 }$$

From the elliptic expansion of true anomaly in terms of mean anomaly:

$$\begin{equation} \theta = 2e\sin M + \mathcal {A}\sin M + \mathcal {O}(e^{2}), \end{equation} \tag{ 4 }$$

where e is Phobos's orbital eccentricity.

In our numerical integration, we compute the force on the planet exerted by Phobos based on the potential (Equation (1)) and the libration angle (Equation (4)):

$$\begin{equation} {\bf F}_{0} = \frac{3}{2}\,\mu _{1}\left(\frac{R^{2}}{\rho ^{4}}\right)[(J_{2}+6\,C_{22}\cos 2\theta)\,\hat{\mbox{\boldmath $\rho $}}_{} +4\,C_{22}\sin 2\theta \,\hat{\bf t}_{}], \end{equation} \tag{ 5 }$$

where $\hat{\mbox{\boldmath $\rho $}}_{}$ is the unit vector directed from Mars toward Phobos and $\hat{\bf t}_{}$ is the unit vector in the Phobos orbit plane normal to $\hat{\mbox{\boldmath $\rho $}}_{}$ and in the direction of Phobos's orbital motion. The reactive force acting on Phobos is

$$\begin{equation} {\bf F}_{1} = -\!\left(\frac{\mu _{0}}{\mu _{1}}\right){\bf F}_{0} \end{equation} \tag{ 6 }$$

with μ₀ being the GM of Mars. This force introduces both secular and periodic perturbations into Phobos's orbit.

We can ascertain the nature of the perturbations by applying Lagrange's planetary equations to disturbing function, Equation (1), to obtain approximate analytical expressions for the perturbations in Phobos's orbital elements:

$$\begin{eqnarray} \Delta \,a &=& 3\,a\left(\frac{R}{a}\right)^{2}(J_{2}+6\,C_{22}) \,e\cos M, \end{eqnarray}\vspace*{4pt} \tag{ 7 }$$

$$\begin{eqnarray} \Delta \lambda &=& 3\left(\frac{R}{a}\right)^{2}(J_{2}+6\,C_{22})\,n\,t +\frac{21}{4}\left(\frac{R}{a}\right)^{2}\nonumber\\ &&\times(J_{2}+6\,C_{22}) \,e\sin M, \end{eqnarray}\vspace*{-10pt} \tag{ 8 }$$

$$\begin{eqnarray} \Delta \,e &=& \frac{3}{2}\left(\frac{R}{a}\right)^{2}(J_{2}+6\,C_{22}) \cos M, \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} \Delta \varpi &=& \frac{3}{2}\left(\frac{R}{a}\right)^{2}\left[J_{2} -2\,C_{22}\left(5+\frac{4\mathcal {A}}{e}\right)\right]\,n\,t \nonumber\\ && +\frac{3}{2}\left(\frac{R}{a}\right)^{2} \frac{(J_{2}+6\,C_{22})}{e}\sin M, \end{eqnarray} \tag{ 10 }$$

where a is the semi-major axis, λ is the mean longitude, ϖ is the longitude of periapsis, n is the mean motion, and t is time from epoch. Because our force has no component normal to the orbit plane, it has no effect on the orbital inclination or longitude of ascending node. Note that only the ϖ secular rate depends on the forced libration. We find that the amplitudes of the periodic terms for a and λ are sub-meter; the e amplitude is about 4 m, and that for ϖ is about 270 m. The secular rates are about 0.35 deg yr⁻¹ and 0.19 deg yr⁻¹ for the mean longitude and periapsis longitude, respectively.

Table 4 contains our estimated Phobos-forced libration amplitude. For comparison purposes, the table includes the libration amplitude observed optically from Viking 1 images by Duxbury & Callahan (1989) and the amplitude and periapsis rate from the analytical work of Borderies & Yoder (1990). The agreement among the amplitudes lends credence to our result. Yet, we emphasize that our determination of $\mathcal {A}$ is indirect, dependent upon our total force model and the postulated Phobos gravity field. If we assume, as suggested by Equations (7)–(10), that the only observable effect of the libration on the orbit is its contribution to the periapsis precession, then we are implicitly finding $\mathcal {A}$ from the difference between the total observed precession and that produced by the Mars gravity field and the solar and planetary perturbations.

Table 4. Phobos Figure Parameters

Reference	$\mathcal {A}$	$\dot{\varpi }$
	(deg)	(deg yr−1)
Duxbury & Callahan (1989)	0.81± 0.5
Borderies & Yoder (1990)	1.19	−0.18
Current	1.03± 0.22	−0.19

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Equation 10 also provides some insight into the difficulty in determining the Phobos gravitational harmonics from the data fit. Again, if we assume that the only observable effect of the figure acceleration is the periapsis precession, it is clear that only one of the three parameters from $J_{2},C_{22}$, and $\mathcal {A}$ can be independently estimated from the observed $\dot{\varpi }$.

By analyzing the observed longitudes of Phobos between 1877 and 1941, Sharpless (1945) discovered the secular acceleration in its mean longitude. All of the theories of the satellite's motion developed subsequently have included the acceleration. Table 5 provides the accelerations, s, from a sampling of the theories which have been fit to observations.

Table 5. Phobos Secular Acceleration and Mars Tidal Parameters

Reference	s × 10−3	κ2	Q	γ
	(deg yr−2)			(deg)
Sharpless (1945)	1.882± 0.171
Shor (1975)	1.427± 0.147
Sinclair (1978)	1.326± 0.118
Jacobson et al. (1989)	1.249± 0.018
Chapront-Touzé (1990)	1.270± 0.008
Emelyanov et al. (1993)	1.290± 0.010
Bills et al. (2005)	1.367± 0.006	0.163	85.6± 0.4	03346± 00014
Rainey & Aharonson (2006)	1.334± 0.006	0.153	78.6± 0.8	03645± 00039
Lainey et al. (2007)	1.270± 0.015	0.152	79.9± 0.7	03585± 00031
Current	1.270± 0.003	0.152	82.8± 0.2	03458± 00009

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It is generally agreed that the cause of the acceleration is the force due to the tide raised on Mars by Phobos. Bills et al. (2005) deduced the acceleration from fit of a quadratic polynomial to the observed Phobos longitudes over the 1877–2004 time frame and computed the tidal lag angle, γ, in an analytical model of a tidally induced acceleration (Redmond & Fish 1964). The lag angle is the angle between the Mars centered position of the tidal bulge and the position at which the tide acts. The tidal quality factor, Q, is related to the lag: Q = cot 2γ. Rainey & Aharonson (2006) extended the work of Bills et al. by incorporating high accuracy observations of the Phobos shadow on Mars made with the Mars Orbiter Camera on MGS. The accelerations and tide parameters from these two analyses appear in Table 5.

In our numerical integration, we modeled the tide raised by Phobos at position ρ* relative to Mars, the position of the tidal bulge, with a second-order potential that acts at position ρ (Mignard 1979):

$$\begin{equation} T = \kappa _{2}\frac{\mu _{1}}{R_{e}}\left(\frac{R_{e}}{\rho }\right)^{3} \left(\frac{R_{e}}{\rho ^{\ast }}\right)^{3} {P}_{2}\left(\cos \gamma \right), \end{equation} \tag{ 11 }$$

where

κ₂ = the Love number of Mars,

μ₁ = GM of Phobos,

R_e = the equatorial radius of Mars,

ρ = the magnitude of $\mbox{\boldmath $\rho $}$,

ρ* = the magnitude of $\mbox{\boldmath $\rho $}^{\ast }$,

P₂ = the Legendre polynomial of degree 2, and

γ = the angle between $\mbox{\boldmath $\rho $}$ and $\mbox{\boldmath $\rho $}^{\ast }$.

Lainey et al. (2007) employed a similar model in their integration. Table 5 gives our results together with theirs; in our fit we estimated γ whereas they estimated Q. The accelerations match but we differ in the tidal parameters primarily because we used different values for μ₁.

There is actually pretty good agreement among the accelerations estimated in data fits with both theories and integrations; the result from the high precision theory of Chapront-Touzé (1990) is identical to that from the integrations. The accelerations found from examining only the longitudes (Sharpless 1945; Bills et al. 2005; Rainey & Aharonson 2006) are all larger than the accelerations from the data fits.