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Iorio's"high-precision measurement"of frame-dragging with the Mars Global Surveyor

In two analyses of orbital data from the Mars Global Surveyor spacecraft, Iorio [1,2] has claimed confirmation of the frame-dragging effect predicted by general relativity. Initially to an accuracy of 6%, and now 0.5%, exceeding the expected accuracy of NASA's Gravity Probe B. It is shown his results come from misinterpreting the MGS data and then altering a key time period.


Introduction
The Mars Global Surveyor (MGS) reached Mars in September 1997. While the main mission was imaging and mapping its surface, its magnetic and gravitational fields were also mapped. The latter began after aerobraking in the Mars atmosphere put the probe in an approximately circular polar orbit, at an altitude of 400 km, in February 1999. The gravitational field data were obtained by observing its orbital motion, via line-of-sight microwave Doppler and range measurements between the probe and Earth stations.
As first described by Lens and Thirring [3], general relativity predicts a rotating massive body would cause a rotational dragging of nearby space-time. In two recent papers, Iorio [1,2] argues this subtle effect can be detected in the MGS orbit. Previous measurements have been reported by Ciufolini and coauthors [4,5], using the Earth-orbiting LAGEOS and LAGEOS II satellites. In that case, the Lens-Thirring effect tends to be obscured by much larger ones, such as those of Earth and ocean tides, and the accuracy of those results has been questioned [6,7].
A definitive finding is expected by year's end, with the announcement of results from NASA's Gravity Probe B. That experiment is an Earth-orbiting gyroscope, designed specifically to measure Lense-Thirring precession, to an accuracy of 1%. Iorio asserts he has already achieved a measurement, first to 6% and now 0.5% accuracy. This has been well publicized. (For example, see [8].) Here we will review the methods and calculations used to arrive at this extraordinary claim.

Analysis of the MGS Orbital Data
Iorio's Mars Global Surveyor papers [1,2] are based on analysis of the MGS orbit by Konopliv et al. [9] Discussing their data, he writes in the first paper: Here we are interested in particular in the out-of-plane portion of its orbit. The root-sum-square out-of-plane residuals over a five-year time interval spanning from 10 February 2000 to 14 January 2005 are shown in figure 1. They have been determined in a Mars-centered coordinate system. . . Due to improved modelling (orientation, gravity, angular momentum wheel de-saturations, and atmospheric drag), the average of such residuals amounts to 1.6 m.
This result can be well explained with the action of the gravitomagnetic field of Mars on the orbit of MGS.
The figure Iorio is referring to was reproduced from Konopliv et al. The original is shown as Figure 1 here. (Unlike Iorio's version, it shows six years of data.) To compare the average residual error in the MGS orbit with the action of the gravitomagnetic field of Mars, Iorio starts with the gravitomagnetic Lens-Thirring precession of a satellite orbit due a rotating central body: (See Ciufolini and Wheeler [10] for a derivation.)Ω LT refers to the secular rate of change in the longitude of the orbit's ascending node, G is the gravitational constant, J the rotating body's angular momentum, c the speed of light, a and e are the orbit's semi-major axis and eccentricity. Referencing Christodoulidis et al. [11], he finds that a change in Ω LT can be expressed as change in the position of the orbit's ascending node N LT as where i is the orbital plane's inclination with respect to the central body's equatorial plane. With ∆P representing a time interval, he combines these expressions to get The factor of 2 in Eq. (1) is lost because ∆N LT represents the average change in position of the ascending node over this time interval. The result from this equation is compared in both papers to the 1.6 meter outof-plane residual orbit error from Konopliv et al. The parameter values he uses are the same in both cases, except for a J which increases slightly, and the values for ∆P , which change from 5 years to 5 years and two months. In his first paper, he gets a predicted 1.5 meter average Thirring precession of the ascending node, which he notes is within 6% of the value from Konopliv et al.
Instead of meters, the result of the second paper is presented as a ratio where the numerator refers to the 1.6 meter residual from Konopliv et al. The predicted Lens-Thirring precession, in the denominator, is 1.610 in this case. Of the uncertainty on the right, he attributes ±0.0052 to uncertainty in the Mars angular momentum J, ±0.0001 to uncertainty in the constant G, and none to the residual ∆N res from observations of the MGS trajectory. What information is actually shown in Figure 1 from Konopliv et al.?
The number of data points is related to a spacecraft trajectory model generated by the JPL Orbit Determination Program [12]. While data from the entire mission is processed to give a global best fit, the output is a sequence of separate trajectory segments, "data arcs," 4 to 6 (Earth) days in length. Each plotted datum point corresponds to the intersection of two data arcs.
Ideally, the modeled segments would fit together as a smooth, continuous trajectory. Since the modeling cannot be perfect, there are mismatches. The orbital period of MGS is 2 hours, so a single data arc consists of many orbits. As Konopliv et al. [9] describe, to gauge and minimize mismatches between successive data arcs, each is extended to overlap the next by a full orbit. The RMS difference in the spacecraft positions is then computed over that orbit. (Iorio refers to these as root-sum-square residuals, although RMS indicates root-mean-square.) Figure 1: RMS orbit overlap differences from Konopliv et al [9]. Reprinted with permission from Elsevier. These position differences are taken in three directions: radial, along-track (the direction the spacecraft is moving), and normal to the orbit plane. Konopliv et al. plot these residual modeling errors in a three-part figure. Part (c), the "normal orbit difference," is the basis of Iorio's analysis, and is included above. The original point made was that, as modeling of the Mars gravity field and various perturbations have improved, the errors have declined. In this case, the normal average orbit error has decreased from a previous 3 to 1.6 meters.
This does not suggest a net precession of the orbit averaging 1.6 meters over the duration of the mission (∆P ). It refers to the average mismatch between successive data arcs. Over the complete trajectory, the underlying directed errors might cancel, or accumulate to a large number of meters. Even individually, the values do not specify orbital precessions.
The normal mismatch between overlapped orbits can be treated as having two parts: an angular tilt of the orbit plane around some axis through the Mars center of mass, plus a normal translation of the orbit plane. (The latter effect would not be expected in a rotationally symmetric system. For a planet with an irregular gravitational field, it may be significant.) Alone, a 1 meter normal translation of the orbit would give a 1 meter RMS difference, with no angular precession of the orbit.
Assuming the normal orbit difference is only due to Lense-Thirring precession, there would be no translation, and the corresponding orbital tilt would be exactly around the Mars rotation axis. In this case, the overlapped orbits would cross near the poles and separate at the Mars equatorial plane. If the ascending node positions differ there by 1 meter, the resulting RMS residual will be less, still not corresponding to what is assumed in Iorios calculation. Also, these numbers have no sign to indicate a direction of precession.
Moreover, there is no reason to expect the value 1.6 is unique. Lemoine et al. [13] and Yaun et al. [14] describe repeated decreases in the MGS orbit residuals as the accuracy of Mars gravity maps and perturbation modeling have improved. Nowhere in Konopliv et al. is it suggested that the present 1.6 m normal average orbit error could not be reduced by further refinement of the existing orbit model.
Another crucial element of Iorio's analysis is his choice of the time period ∆P . According to him, the out-of-plane orbit error corresponds to a cumulative precession, increasing with time. The 1.6 m error from Konopliv et al. is based on 6 years of data, shown in Figure 1. Without explanation, Iorio's first paper takes the same error to apply to a 5-year period. (His figure omits the data prior to 2000 to agree with that.) In the second paper, he uses a ∆P equal to 5 years + 2 months, and obtains an order of magnitude better agreement between general relativity and experiment. This is possible because, while changing ∆P and with it the predicted error, he compares that to the same 1.6 m measured error once again. This time he notes By using a time span ∆P of slightly more than 5 years (14 November 1999−14 January 2005, so to remove the first months of the mission more affected by orbital maneuvers and non-gravitational pertur- Two statistical tests are then used to show it is unlikely the close correspondence between his predicted value and that observed is due to chance. (This conclusion seems justified.) Stressing the objectivity of his findings, he also writes Another important remark concerning our measurement is the following one: the interpretation of the MGS residuals presented in this paper and their production from the collected data were made by distinct and independent subjects in different times; moreover, the authors of [9] had not the Lense-Thirring effect in mind at all when they performed their analysis. Thus, the possibility of any sort of (un)conscious gimmicks, tricks or tweaking the data because the expected answer was known in advance [15] is a priori excluded in this test.
However, his results come from misinterpreting the MGS data, and from his own values of ∆P -twice altered in a way which gives closer agreement with general relativity.

Discussion
It would be surprising if an accurate test of Lens-Thirring precession (or any test of it) could be obtained from the Mars Global Surveyor trajectory. Unlike some spacecraft, it was not designed for measurement of its undisturbed inertial motion. That was possible, for example, with the Pioneer 10 and 11 space probes [16]. Those encountered no planetary atmospheres, and were spin-stabilized, such that frequent use of thrusters was not required. In this case, AMD (angular momentum wheel desaturation) thrusters were fired 4 or 5 times per day until September 2001, when that was reduced to 1 or 2. Velocity uncertainties due to imprecision of the thrust are partially corrected from observations of the spacecraft motion, but that is limited by the one-dimensional nature of the Doppler velocity measurements.
From Konopliv et al.: "Normal orbit error is the greatest, and along-track is the least when the orbit plane is near edge-on as viewed from Earth (e.g., January 2001). For this geometry, the signature in the Doppler (which measures the velocity of the spacecraft in the Earth-Mars direction) is greatest for motion in the orbit plane, but is minimal for motion normal to the orbit plane." Referring to Figure 1, note the dips in the radial and along-track differences centered around January 2001, and the corresponding large hump in the normal orbit difference, where it increases by a factor of ten. This obviously contributes to the 1.6 m average normal orbit difference. We could agree with with Konopliv et al. that this is due to measurement error. Certainly it cannot be attributed to a larger general relativistic precession occurring at that time.
Iorio's second paper states his value of ∆P was chosen "so to remove the first months of the mission more affected by orbital maneuvers and non-gravitational perturbations." However, there was no change in orbital maneuvers which would justify a time period beginning in November 1999. Also, Konopliv et al. point out that noise in the Doppler velocity signal was least during the first one and three quarters months, referred to as the "Gravity Calibration Orbit." This preceded deployment of the large high-gain antenna, which increased atmospheric drag.
At the relatively low altitude of MGS, the Martian atmosphere is a significant factor. From the spacecraft's asymmetric shape, drag due to its forward motion would be expected to cause out-of-plane forces. For a satellite in polar orbit, rotation of the upper atmosphere would also tend to produce a force causing an orbital precession. Solar heating of the spacecraft on emergence from the Mars shadow may be an additional factor.
None of the possible errors due to these or other orbital perturbations are estimated by Iorio, or included in his calculations. Following initial publication of this review on arXiv, another by Sindoni, Paris and Ialongo [17] has found the uncertainty of the orbit's precession exceeds that claimed in Iorios second paper by at least a factor of 10000. Also see Felici [18].

Conclusions
Iorio makes the hypothesis that a certain measure of the error in the modeled Mars Global Surveyor orbit is due to the Lense-Thirring precession from general relativity. He assumes this measure is precisley 1.6 meters. This is shown to equal general relativity's prediction, within a factor of 0.9937, and with an uncertainty of ±0.0053. However, close agreement was obtained by calculating the predicted precession for a different time period than the actual period of observations. Also, where he assumes 1.6 meters represents the average of a cumulative effect arising over 5 or 6 years, it actually characterizes an average RMS orbital mismatch arising over two "data arcs" -with a total span in the range of 8 to 12 days. The effects are generally not cumulative. Finally, this quantity is not a specific measure of movement of the orbit's ascending node, as assumed in Iorio's calculations.
Other approaches to measuring the Lens-Thirring effect are promising. In addition to the pending result from Gravity Probe B, a new experiment has been proposed by Turyshev et al. [19] This would observe gravitational bending of laser light by the Sun, using two small, relatively inexpensive satellites positioned on the far side of the Sun, and the International Space Station. Second-order bending would be measured for the first time, as well as that due to frame dragging caused by the Sun's rotation. The latter would be measured to 1% accuracy.