The communication antennas of the DSN complex that are used to exchange data with the Pioneer spacecraft are located on the Earth’s surface. This introduces many corrections to the modeling of the uplinked or downlinked radio signal due to the orbital motion, rotation, internal dynamics and atmosphere of our home planet.
The interplanetary medium in the solar system is dominated by the solar wind, i.e., charged particles originating from the Sun. Although their density is low, the presence of these particles has a noticeable effect on a radio frequency signal, especially when the signal passes relatively close to the Sun.
Delay due to solar plasma is a function of the electron density in the plasma. Although this can vary significantly as a result of solar activity, the propagation delay (in microseconds) can be approximated using the formula :
We write the electron density as a sum of a static, steady-state part, and fluctuation :, except at conjunction, when noise due to the solar corona dominates the Doppler observable. In contrast, the steady-state behavior of the solar corona is well known and can be approximated using the formula [19, 148, 242, 243]:
Using Equation (4.28) in Equation (4.25), we obtain the range model :
The values of the parameters , , and are: , all in meters .
As the radio signal to or from the spacecraft travels through the Earth’s ionosphere, it suffers an additional propagation delay due to the presence of charged particles. This delay can be modeled as [208, 324]6, while is well approximated by the formula × 1016 electrons during the day at various stages during the solar cycle.
Chao (; see also [121, 382, 166, 205, 330]) estimates the delay due to signal propagation through the troposphere using the following formula:
Unfortunately, historical weather data going back over 30 years may not be available for most DSN stations. In the absence of such data, C.B. Markwardt suggests that seasonal weather data or historical weather data from nearby weather stations can be used to achieve good modeling accuracy.23
The radio signal emited by the DSN and the radio signal returned by the Pioneer 10 and 11 spacecraft are circularly polarized. The spacecraft themselves are spinning, and the spin axis coincides with the axis of the HGA. Therefore, every revolution of the spacecraft adds a cycle to both the radio signal received by, and that transmitted by the spacecraft.
At a nominal rate of 4.8 revolutions per minute, the spacecraft spin adds 0.08 Hz to the radio signal frequency in each direction.
The sign of the spin contribution to the spacecraft frequency depends on whether or not the radio signal is left or right circularly polarized, and the direction of the spacecraft’s rotation.
The rotation of the spacecraft is clockwise  as viewed from a direction behind the spacecraft, facing towards the Earth. This implies that the spacecraft spin would contribute to the frequency of a right circularly polarized (as seen from the transmitter) signal’s frequency with a positive sign. The assumption that the DSN signal is right circularly polarized is consistent with the explanation provided in . This interpretation of the spacecraft’s spin in relation to the radio signal agrees with what one finds when comparing orbit data files with or without previously applied spin correction.
The total amount of spin correction, therefore, must be written as
Accurate estimation of the amount of time it takes for a signal to travel between a DSN station and a distant spacecraft, and the frequency shift due to the relative motion of these, requires precise knowledge of the position and velocity of not just the spacecraft itself, but also of any ground stations participating in the communication.
DSN transmitting and receiving stations are located on the surface of the Earth. Therefore, their coordinates in a solar system barycentric frame of reference are determined primarily by the orbital motion, rotation, precession and nutation of the Earth.
In addition to these motions of the Earth, station locations also change relative to a geocentric frame of reference due to tidal effects and continental drift.
Information about station locations is readily available for stations presently in operation; however, for stations that are no longer operating, or for stations that have been relocated, it is somewhat more difficult to obtain (see Section 3.1.2).
The transformation of station coordinates from a terrestrial reference frame, such as ITRF93, to a celestial (solar system barycentric) reference frame can be readily accomplished using publicly available algorithms or software libraries, such as NASA’s SPICE library24 .
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