The Constancy of the Pioneer Anomalous Acceleration
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A&A 463, 393–397 (2007) Astronomy DOI: 10.1051/0004-6361:20065906 & c ESO 2007 Astrophysics The constancy of the Pioneer anomalous acceleration Ø. Olsen Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029 Blindern, 0315 Oslo, Norway e-mail: [email protected] Received 26 June 2006 / Accepted 8 November 2006 ABSTRACT Aims. The temporal evolution of the Pioneer anomalous acceleration is studied to determine its correlation with solar activity. Methods. Bayesian Weighted Least Squares algorithms and Kalman filters are used to study the temporal evolution of the Pioneer anomalous acceleration. Doppler data for Pioneer 10 and 11 spanning respectively 11 and 3.5 years were used in this study. Results. The Doppler data itself can not be used to distinguish between a constant acceleration model and acceleration proportional to the remaining plutonium in the radioisotope thermoelectric generators. There are significant short term variations in the anomalous acceleration. These are shown to be consistent with radio plasma delay from coronal mass ejections. Key words. celestial mechanics – ephemerides 1. Introduction transmission time at the receiving and transmitting electronics. The integration limits t and t are the beginning and end of The Pioneer 10 and 11 missions were the two first spacecraft to 3s 3e 1 each count interval at the receiving station, while t1s and t1e are explore the outer Solar System. Anderson et al. (2002) found the start and end times of the corresponding interval at the uplink a small unmodeled acceleration acting on the spacecraft. The station. t and t can be found from t and t and the precision magnitude of the acceleration is approximately 8 × 10−8 cm/s2 1s 1e 3s 3e round-trip light time solutions ρe and ρs. The precision round- with a direction towards the inner Solar System. Explanations of trip light times were computed from this anomaly range from the new physics to overlooked conven- tional forces. Scheffer (2003) argues for non-isotropic radiation r r ρ = 23 + RLT + 12 + RLT of spacecraft heat as the most likely candidate. In this Paper I c 23 c 12 study the temporal variations of the anomalous acceleration to −∆Tt3 − ∆Tt1 see if any model can be excluded. 1 PA together with its references provide comprehensive de- + ∆Aρ(t3) +∆SCρ23 +∆Aρ(t1) +∆SCρ12 , (2) scriptions of the Pioneer spacecraft and the Deep Space Network c (DSN) communications system. This will not be repeated in this where RLTij is the Shapiro delay on the upleg and downleg light paper. time solutions. r23 is the distance between receiver at reception time and the spacecraft at retransmission time. Similarily, r12 2. Analysis of the data is the distance between transmitter at transmission time and the spacecraft at retransmission time. c is the speed of light. ∆Aρ(ti) Moyer (2000) provides the formulation for observed and com- are the off-axis antenna corrections to the receiving and trans- puted values of DSN data types for navigation. Only a cursory mitting station on Earth. The values used to calculate these cor- description for doppler data is provided in this paper as the Orbit rections where taken from Imbriale (2002). ∆SCρ3 and ∆SCρ1 are Data File (ODF) tracking data consists of time series of observed the solar corona corrections on the down and up leg. The Cassini doppler counts, which can be converted to time series of doppler model was used for this study, but as pointed out by PA, the net frequencies with given count intervals, TC. effect of the solar corona is small and can ultimately be ignored. Moyer designates the transmission time from Earth as t1,the The model itself is given by PA. ∆T3 and ∆T1 are the differences epoch of retransmission as t2 and the reception time as t3.At between Coordinated Universal Time (UTC) and Barycentric each of these epochs the Solar System Barycentric (SSB) po- Dynamical Time (TDB) at transmission and reception time. sition of the uplink station r1, spacecraft r2 and downlink sta- Troposphere and ionosphere corrections were not included tion r3 must be calculated. The range rij is defined as the mag- in the light time solution, but added as a media correction to the nitude of the vector r j − ri. Computed values for two-way and Doppler observable. Media corrections for two-way and three- three-way doppler observables are then found from way computed Doppler observables are given by t3e t1e M2R M2 F , = f (t )dt − f (t )dt , (1) M2 2 3 T 3 3 T 1 1 ∆F , = ( f (t )∆ρ − f (t )∆ρ ) . (3) TC t TC t 2 3 T 1e e T 1s s 3s 1s TC M = M = / S where 2R 2 240 221 for -band communication. ∆ρ ∆ρ f (t )and f (t ) are the transmitter frequencies at reception and e and s are the correction to the precision round-trip light T 3 T 1 times at the end and beginning of the transmission interval. The 1 Herafter denoted as PA. frequencies at the start and end of the transmission interval must Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20065906 394 Ø. Olsen: The constancy of the Pioneer anomalous acceleration be known. Media corrections are computed both for the up-leg To study the variation of the anomalous acceleration, I imple- and down-leg part of the light time solution. mented a Kalman filter with stochastical variables. The anoma- Finally, partials of the computed Doppler observables with lous acceleration was modeled as a first order Markow process respect to solve-for parameters q are required: with colored noise: ∂F , M ∂ρ ∂ρ p + = m(t + , t )p + e (5) 2 3 = 2 f (t ) e − f (t ) s · (4) k 1 k 1 k k k ∂q T T 1e ∂q T 1s ∂q C and − − /τ , = (tk+1 tk) , 2.1. Relativistic equations of motion m(tk+1 tk) e (6) The spacecraft ephemeris is generated by numerically integrat- where τ is the correlation time. tk+1 and tk are the time-tags ing the equations of motion using the Bulirsh-Stoer method of observation number k and k + 1. A correlation time of zero (Press et al. 1996). The equations of motion were represented corresponds to a white noise process while an infinite correla- by the β-andγ-parameterized post-Newtonian (PPN) approx- tion corresponds to a random walk process. The evolution of the imation formulated in the solar system barycentric frame of stochastic parameter is dependent on the correlation time and the reference (Moyer 2000). General relativity corresponds to β = standard deviation, σe, of the noise vector ek. Some care must γ = 1 which were chosen for this study. Relativistic equations be taken when selecting σe. A too large value of σe allows ab- of motion are not necessary to analyze the Pioneer anomalous sorbing all of the doppler-noise into the anomalous acceleration, acceleration as demonstrated by Markwardt (2002). They are while a too small value forces the stochastic parameter to re- more than four orders of magnitude smaller than the anomalous main constant. Guided by the results of PA, σe = a0τe was cho- acceleration. sen. This preserves short-term structures and the general noise A natural choice for this study is the International Celestial level of the doppler data. The estimate of the anomalous accel- Reference Frame (ICRF) with the SSB as origin. The coordi- eration at a given time tm is based only on the preceding data- nates of the uplink station, spacecraft and downlink station (r1, points, and is therefore not optimal estimates. Hence, after for- r2 and r3) must all be expressed in the same coordinate system. ward processing of the data-sets, the results were smoothed with r2 was interpolated from the spacecraft ephemeris and therefore the Rach-Tung-Striebel (RTS) algorithm (Rauch et al. 1965). already in the desired frame of reference. The station coordi- nates had to be transformed from the International Terrestrial Reference Frame (ITRF) to the ICRF and then transformed 3. Results and discussion from the geocentric to barycentric coordinates. The International The dataset spans the time-interval from January 1987 to Earth Rotation and Reference Systems Service (IERS) 2003 con- July 1998, which is the same interval analyzed in PA. I report vention (McCarthy & Petit 2003) and Moyer (2000) covers these best fit values corresponding to each of the three intervals de- transformations. fined by PA. Interval I spans January 1st 1987 to 17 July 1990, The coordinates of the Sun, the planets and the Moon with Interval II spans July 17th 1990 to July 12th 1992 and Interval III respect to the SSB were interpolated from the Jet Propulsion continues up to July 21st 1998. Laboratory (JPL) DE405 planetary ephemeris, which uses the The parameters included in the fitting model were the ini- ICRF. The time argument of DE405 is TDB (Standish 1998), tial SSB position and velocity, a constant anomalous acceler- which was used while integrating the equations of motion. ation and instantaneous velocity increments along the Earth- spacecraft line for each maneuver. Because of the Pioneer spin, 2.2. Parameter estimation algorithms these maneuvers are only important along the Earth-spacecraft line, and I chose to use the same model as Markwardt (2002) The parameters were estimated using an ordinary Bayesian for the maneuvers. PA models the maneuvers as instantaneous Weighted Least Squares (BWLSQ) algorithm, although I ini- velocity increments along the three spacecraft axes. The timing tially used a Kalman filter to check the numerical accuracy and of each maneuver was provided by Slava Turyshev.