### 4.5 Effects on the radio signal

The radio signal to or from the Pioneer spacecraft travels several billion kilometers in interplanetary space. Unsurprisingly, the interplanetary environment, notably charged particles emitted by the Sun and the gravitational fields in the solar system all affect the length of the path that the radio signal travels and its frequency.

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.

#### 4.5.1 Plasma in the solar corona and weighting

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 [148]:

where is the signal frequency, is is the critical plasma density for frequency that is given by
and is the electron density as a function of time and position , which is integrated along the propagation path between the spacecraft and the Earth.

We write the electron density as a sum of a static, steady-state part, and fluctuation  [392]:

The second term, which is difficult to quantify, has only a small effect on the Doppler observable [392], 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 [19148242243]:
where is the solar radius, and is the distance from the Sun along the propagation path.

Using Equation (4.28) in Equation (4.25), we obtain the range model [27]:

where is the reference frequency used for the analysis of Pioneer 10, is the impact parameter with respect to the Sun, and is a light-time correction factor, which is given for distant spacecraft as
where and are the heliocentric radial distances to the target and to the Earth, respectively. The sign of the solar corona range correction is negative for Doppler measurements (positive for range).

The values of the parameters , , and are: , all in meters [27].

#### 4.5.2 Effects of the ionosphere

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 [208324]

where is the antenna elevation, is the atmospheric refractivity index, is the ionospheric refractivity index, and is the height of the ionosphere. can be approximated at 106, while is well approximated by the formula
where is the electron density and is the signal frequency in Hz. We introduce the total electron content,
, which allows us to express the propagation delay in the form,
with . For European latitudes, the total electron content may vary from very few electrons at night to (20 –,100) × 1016 electrons during the day at various stages during the solar cycle.

#### 4.5.3 Effects of the troposphere

Chao ([353]; see also [121382166205330]) estimates the delay due to signal propagation through the troposphere using the following formula:

where is the additional propagation path, is the elevation angle, and and are coefficients defined as

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.

#### 4.5.4 The effect of spin

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 [292] 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 [240]. 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

where is the angular velocity of the spacecraft, and we accounted for the Pioneer communication system turnaround ratio of 240/221.

#### 4.5.5 Station locations

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 library [4].