SPACE NAVIGATION AND FLIGHT DYNAMICS

1-MAY-2014

MESSENGER Navigation Orbit Reconstruction 1 (ORECON1)

Prepared by: Tony Taylor

This document describes the reconstruction, by the MESSENGER Navigation team, of the trajectory/orbit of the MESSENGER spacecraft from 2011/01/01 to 2012/04/26. The name given by the Navigation team for this reconstruction is ORECON1 (Orbit RECONstruction 1). Delivery Files The following files were delivered to the standard delivery location for reconstructed spacecraft ephemerides.

ORECON1.bsp

This SPICE SPK file covers the approach trajectory extending from 2011/01/01 through Mercury Orbit Insertion (MOI) on 2011/03/18 (UTC), continuing in the orbit phase until 2012/04/26 04:00:00 SCET, TDB. The file also contains the DE405 planetary ephemerides for all bodies except the Mercury and Earth barycenters. Earth and Mercury barycenter ephemerides were solved for, and these estimated ephemerides (SPK IDs 1 and 3) replace the corresponding ephemerides from DE405.

ORECON1_Sigmas.m This file, in Matlab script format, contains the 1-sigma solution uncertainties corresponding to the trajectory/orbit data of ORECON1.bsp. A detailed description of the tabular data and coordinate system is given in the header.

ORECON1_Apses.m This file, also in Matlab format, contains an array of all periapsis times in calendar string format, and another array of all apoapsis times. These were generated from ORECON1.bsp.

Comparison to Trajectory Files Produced during Operations This post-operations reconstruction benefits from a number of modeling improvements:

Doppler data is weighted more strongly, for the equivalent of 0.2 mm/s 1-sigma noise (normalized to one minute) rather than the 0.5 mm/s used by operations.

Uses five-second Doppler data compression within +/– 40 minutes of each periapsis for higher resolution than the one minute compression used throughout orbits in operations.

Corrects a deficiency in the Mercury thermal model (input to the Planet Radiation Pressure model. This was also corrected for operations in August of 2012.)

Estimates both Mercury and Earth ephemerides, eliminating a problem induced by estimating Mercury but “Considering” Earth.

May 1, 2014 –2–

Uses smaller time resolution of 6 seconds for spacecraft attitude modeling than the 60 seconds used by operations.

Uses DE405 as the baseline planetary ephemeris to avoid subtle reuse of the same radiometric data implicit in the DE423 ephemeris produced by JPL using MESSENGER data. (Earth and Mercury barycenter ephemerides were solved for and updated in the delivered SPK file).

Tweaks a priori constraints on Planet Radiation Pressure and Solar Radiation Pressure parameters to ease the process of iterating solutions to linearity.

Changes a priori filter information to place more emphasis on the earlier flyby determinations of Mercury GM (since the flybys were more sensitive to GM than was the case for the orbit phase).

Loosely ties the epoch state of each weekly data arc to the end of the preceding arc, using values and sigmas from the preceding arc as a priori information for the new arc.

Figures 1–3 show the radial, transverse, and normal (RTN) direction differences between this reconstructed trajectory and the current Mission Design reference trajectory (which is pieced together with trajectory segments from operations deliveries by the navigation team, such as OD204, OD205, and so forth). Radial differences are quite small for the trajectories at both apses (Figures 2 and 3, standard deviation of 13 meters) but larger (1.2 km) for the final approach in Figure 1. Figure 4 shows excerpts from the file containing periapsis and apoapsis times.

Figure 1: Trajectory differences in last part of cruise, in RTN system (R, HxR, H).

May 1, 2014 –3–

Figure 2: Trajectory differences at periapses, in RTN system (R, HxR, H).

Figure 3: Trajectory differences at apoapses, in RTN system (R, HxR, H).

May 1, 2014 –4–

Although crowded and difficult to read, the date/time labels for Figures 2 and 3 are given as they are to show the end times of the individual trajectories that were concatenated together to produce the reconstruction. The labels are placed at the date and hour of the Data Cut Off (DCO) times of each data arc. Each data arc was constructed to start at the nearest round hour before an apoapsis, and end at the nearest round hour (the DCO) before an apoapsis about a week later. The beginning and ending apoapses were chosen to precede the first periapsis on Thursday of each week, and follow the last periapsis on Wednesday of the following week. Thus the arcs, although only approximately a week long, are tailored to include exactly a week’s worth of periapses. This scheme assured that arcs never began or ended at an awkward place, e.g., close to periapsis, and that there were never any overlaps in the tracking data.

Figure 4: Excerpts of tabular data from ORECON1_Apses.m containing calendar string times for periapses and apoapses generated from the SPK delivery file.

Orbit Uncertainties Figure 5 gives a sample of the contents of ORECON1_Sigmas.m. Figures 6–8 are plots of data from that file.

%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! % output/ORECON1_Apses.m % Created: 24-Apr-2014 16:39:26 by ApsisTimeGen.m % Spice Files: % /nav/input/leapsec/latest % /nav/msgr/od/work/8_orbit/recon/ORECON1/140424_aht/output/ORECON1.bsp % Begin: 17-MAR-2011 00:00:00. TDB % End : 26-APR-2012 04:00:00. TDB % 827 periapsis times % 827 apoapsis times % Column 1: Times [SCET, TDB] % Column 2: Count % Column 3: Hours since previous % Column 4: Days since first sPerTimes = [ % Periapsis times. '18-MAR-2011 00:51:45.013 TDB' % 1 0.000000 0.000000 '18-MAR-2011 12:53:26.017 TDB' % 2 12.028057 0.501169 '19-MAR-2011 00:57:42.901 TDB' % 3 12.071357 1.004142 '19-MAR-2011 13:02:00.092 TDB' % 4 12.071442 1.507119 . . . . . . . . . . '24-APR-2012 23:09:56.764 TDB' % 824 8.000599 403.929303 '25-APR-2012 07:09:58.894 TDB' % 825 8.000591 404.262661 '25-APR-2012 15:10:01.027 TDB' % 826 8.000593 404.596019 '25-APR-2012 23:10:03.233 TDB' % 827 8.000613 404.929378 ]; sApoTimes = [ % Apoapsis times. '18-MAR-2011 06:51:17.875 TDB' % 1 0.000000 0.000000 '18-MAR-2011 18:55:34.519 TDB' % 2 12.071290 0.502970 '19-MAR-2011 06:59:51.459 TDB' % 3 12.071372 1.005944 '19-MAR-2011 19:04:08.739 TDB' % 4 12.071466 1.508922 . . . . . . . . . . '25-APR-2012 03:09:57.780 TDB' % 824 8.000594 403.846295 '25-APR-2012 11:09:59.904 TDB' % 825 8.000590 404.179653 '25-APR-2012 19:10:02.070 TDB' % 826 8.000602 404.513012 '26-APR-2012 03:10:04.282 TDB' % 827 8.000614 404.846370 ];

May 1, 2014 –5–

The large peaks seen in Figure 7 in transverse (HxR) position in the vicinity of 2011/06/15 and 2011/12/08 are induced by a geometry in which the orbit-normal direction (H) is close to the direction (or anti-direction) to Earth; thus the Doppler is not sensitive to down-track position of the spacecraft. Due to the same geometry, Doppler is very sensitive to out-of-plane motion of the spacecraft, and there are noticeable dips in the H direction errors during these times. Figures 7 and 8 underline the concept that no small set of numbers or simple algorithm can adequately characterize the uncertainties inherent in orbit determination during the orbit phase. The errors vary with location within an orbit, amount of tracking data available in a data arc, multiple adverse geometries that can last from days to weeks, and the time within the one-week solution arcs at which the uncertainties are computed. This is the reason the tabular data are provided. Because of difficulties with the orbit determination in some data arcs (such as unrealistic behavior of solved-for parameters in the Solar Radiation Pressure and Planet Radiation Pressure models, usually hand-in-hand with unexplained signatures in the post-fit Doppler data), an “unconfidence factor” was assigned for some arcs and the corresponding trajectory errors for those arcs were scaled up. Sixteen of these arcs were thus treated with the scale factors seen in Table 1. These scale factors were assigned subjectively, intended to put the troublesome arcs on roughly the same 1-sigma footing as the ones that presented no difficulties. The 1-sigma uncertainties of ORECON1_Sigmas.m are formal values, meaning that they are unaltered (except for the unconfidence scale factors) from the computed values, and as such they are optimistic to an unknown extent. There is no definitive formula for providing realistic sigmas, so orbit uncertainties here are empirically estimated.

May 1, 2014 –6–

Figure 5: Excerpts of tabular data from ORECON1_Sigmas.m containing times, radial distance, transverse velocity, and formal 1-sigma values for position uncertainties relative to Mercury center.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ORECON1_Sigmas.m % 24-Apr-2014 % MAP FRAME: View frame 1 (inertial) MAP CENTER: Mercury % TRANSFORM: Cartesian MAP STATES: Msgr % Column names: % DCO_TIME MAP_TIME R V_HxR SIG(R) SIG(HxR) SIG(H) % Column descriptions: % (Times in seconds past J2000 [SCET, TDB]) % 1: DCO_TIME: Data Cut Off times, each solution % 2: MAP_TIME: Output (computation) times % 3: R: Radial distance [km] % 4: V_HxR: Velocity along HxR direction [km/s] % 4: where H = orbit pole % 5: SIG(R): 1-sigma position along R [km] % 6: SIG(HxR): 1-sigma position along HxR [km] % 7: SIG(H): 1-sigma position along H [km] % 8: MAP_TIME: As a commented time string % Map files and sigma scaling factors are commented % at the beginning of each solution data arc. % % Periapsis/apoapsis output times (corresponding to % minimum/maximum values in the R column) are rounded % to about +/-0.05 seconds. % Other output times are centered in 15 minute steps % for +/- 2 hours around periapses and 1 hour steps % around apoapses (except for the approach solution data % arc ending 2011/03/18). % Sigs = [ % /nav/msgr/od/gravity/2011/110318/121028_aht_v2/output/map5.txt.30 % Sigma scale factor = 1.0 353743200 347115104.980 65940365.433 40.922852 0.211 0.575 3.005 % 2011/01/01 00:51:44.980 TDB 353743200 347201504.980 64693450.040 39.184192 0.225 0.575 3.013 % 2011/01/02 00:51:44.980 TDB 353743200 347287904.980 63435633.930 37.519364 0.240 0.574 3.011 % 2011/01/03 00:51:44.980 TDB 353743200 347374304.980 62170026.658 35.928193 0.255 0.571 3.000 % 2011/01/04 00:51:44.980 TDB . . . . . . . . . . 353743200 353740004.950 16580.012 0.607531 0.005 0.082 0.315 % 2011/03/18 17:06:44.950 TDB 353743200 353740904.980 16873.540 0.596962 0.005 0.083 0.319 % 2011/03/18 17:21:44.980 TDB 353743200 353741805.010 17121.458 0.588319 0.005 0.085 0.321 % 2011/03/18 17:36:45.010 TDB 353743200 353742704.950 17324.839 0.581412 0.005 0.086 0.323 % 2011/03/18 17:51:44.950 TDB% /nav/msgr/od/gravity/2011/110323/130420_aht_v2/output/map5.txt.4 % Sigma scale factor = 1.0 354178800 353746534.520 17708.491 0.568816 0.003 0.038 0.134 % 2011/03/18 18:55:34.520 TDB 354178800 353750134.550 17369.697 0.579910 0.003 0.037 0.125 % 2011/03/18 19:55:34.550 TDB 354178800 353753734.490 16330.941 0.616796 0.003 0.035 0.109 % 2011/03/18 20:55:34.490 TDB 354178800 353757334.520 14516.490 0.693890 0.003 0.032 0.086 % 2011/03/18 21:55:34.520 TDB . . . . . . . . . . 388681200 388674603.240 10367.437 0.958494 0.002 0.031 0.120 % 2012/04/26 01:10:03.240 TDB 388681200 388678204.310 12177.719 0.816010 0.003 0.038 0.149 % 2012/04/26 02:10:04.310 TDB 388681200 388681804.250 12753.340 0.779183 0.003 0.040 0.154 % 2012/04/26 03:10:04.250 TDB ];

May 1, 2014 –7–

Figure 6: RTN uncertainties in final approach phase. (The event near 2011/03/02 is due to a coordinate frame singularity, not a physical event.)

Figure 7: RTN 1-sigma (formal) uncertainties in orbit phase, with periods of poor orbit determination geometry indicated.

May 1, 2014 –8–

Figure 8: Details of 1-sigma uncertainties for data arc beginning 2011/06/23 04:00 and ending 2011/06/30 01:00.

Table 1: Sigma Scale Factors other than 1.0

Sigma Scale Factor

Data Arc Ending

1.5 2011/04/20 23:00

2.0 2011/05/19 03:00

2.0 2011/06/30 01:00

1.5 2011/07/06 23:00

1.5 2011/07/13 20:00

3.0 2011/07/21 05:00

1.5 2011/08/18 03:00

4.0 2011/09/08 02:00

1.5 2011/12/01 04:00

1.5 2011/12/08 03:00

2.0 2011/12/21 21:00

1.5 2012/03/01 02:00

1.5 2012/03/21 22:00

2.0 2012/03/29 04:00

2.0 2012/04/04 22:00

1.5 2012/04/12 05:00

May 1, 2014 –9–

Gravity The gravity solution was more a byproduct of the reconstruction process than a targeted result. The “square root information” arrays were saved for each data arc after the solution was in hand, and they were combined at the end of the process with little to no experimentation along the way. Thus it was reassuring to find that this solution differs little from the navigation gravity solutions that preceded it. The following is the result for seven full inertial rotations of Mercury plus the results for the three flybys. Figure 9 shows solved values and sigmas for the low-order terms from file MNG04_C.m. The full 20-degree solution along with the covariance is given in the file. Figure 10 presents the spectrum of coefficients, where the Kaula constraint was 80e-6/n2, and the other two Kaula schedules are shown for comparison. Figure 11 shows contours for the gravity potential (excluding GM). The contours of Figure 12 of the 1-sigma potential uncertainties verify that the longitude coverage was complete and fairly uniform.

Figure 9: Excerpts from file MNG04_C.m containing solved-for gravity parameters and 1-sigma uncertainties.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MNG04_C.m % MNG04 % MESSENGER Navigation Gravity Field for Mercury % Tony Taylor, KinetX, Inc. % 27-Apr-2014 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % MNG04_C.m % Matlab formatted output from Export2CS % Basis: Export file MNG_AP02_110318_120425_v02_g_E.m % 438 of 438 parameters output. NAMES = { 'GM1' '1C02_00' '1C02_01' ... '1S02_01' '1C02_02' '1S02_02' ... '1C03_00' '1C03_01' '1S03_01' ... '1C03_02' '1S03_02' '1C03_03' ... '1S03_03' '1C04_00' '1C04_01' ... . . . . . VALUES = [ 2.203187230534321e+04 -2.259729482885131e-05 -3.658706279739164e-09 ... -8.664850822505650e-10 1.250884617612498e-05 -4.731820653351541e-08 ... -4.718246766844097e-06 -3.563424565615864e-06 -2.590144696423164e-06 ... 9.757760751865151e-07 -6.779810231490561e-07 6.258472203288019e-07 ... 1.692104977133469e-06 -5.890871045005338e-06 1.618465672157473e-06 ... . . . . . SIGMAS = [ 2.404408518863336e-04 2.771044967699667e-09 8.870926298314602e-10 ... 8.965202578942121e-10 1.459408451981861e-09 1.573422398987087e-09 ... 7.091644718252861e-09 5.566838890515535e-09 6.100991121837986e-09 ... 3.563992132586956e-09 3.753190158600194e-09 1.737261618655086e-09 ... 1.684031620356337e-09 2.330284337863237e-08 1.870783840784416e-08 ... . . . . .

May 1, 2014 –10–

Figure 10: Spectrum of Mercury gravity coefficients for Kaula constraint of 80e-6/n2.

Figure 11: Contours of Mercury’s gravity potential at the surface on a cylindrical projection.

May 1, 2014 –11–

Figure 12: Contours of the gravity potential 1-sigma uncertainties at Mercury’s surface.

Summary Files are delivered for the first orbit-phase reconstruction, covering the interval from 2011/01/01 to 2012/04/26—seven full inertial rotations of the planet during the orbit phase beginning 2011/03/18. The SPICE SPK file contains both the spacecraft ephemeris and the estimated Earth and Mercury barycenter ephemerides, plus the remainder of the DE405 planetary ephemerides over the same time span. Associated files include 1-sigma formal uncertainties for the trajectory, as well as periapsis and apoapsis times.