4.2 Light times and time scales

The complex gravitational environment of the solar system manifests itself not just through its effects on the trajectory of a spacecraft or celestial body: the propagation of electromagnetic signals to or from an observing station on the Earth must also be considered. Additionally, proper timekeeping becomes an issue of significance: clocks that are in relative motion do not tick at the same rate, and changing gravitational potentials may also affect them.

4.2.1 Light time solution

The time it takes for a signal to travel between two locations in space in the gravitational environment of a massive point source with gravitational constant μ can be derived from the PPN metric Equation (4.1View Equation) in the form [240Jump To The Next Citation Point]:

2 t2 − t1 = r12 + (1 + γ)-μ ln r1-+-r2 +-r12-+-(1-+-γ)μ∕c-+ 𝒪 (c−5), (4.4 ) c c3 r1 + r2 − r12 + (1 + γ)μ∕c2
where t 1 refers to the signal transmission time, and t 2 refers to the reception time, r 1,2 represent the distance of the point of transmission and point of reception, respectively, from the massive body, and r12 is the spatial separation of the points of transmission and reception. The terms proportional to μ∕c2 are important only for the Sun and are negligible for all other bodies in the solar system.

4.2.2 Standard time scales

The equations of motion Equation (4.2View Equation) and the light time solution Equation (4.4View Equation) are both written in terms of an independent time variable, which is called the ephemeris time, or ET. Ephemeris time is simply coordinate time in the chosen coordinate frame, such as a solar system barycentric frame. As such, the ephemeris time differs from the standard International Atomic Time (TAI, Temps Atomique International), measured in SI (Système International) seconds relative to a given epoch, namely the beginning of the year 1958.

When a solar system barycentric frame of reference is used to integrate the equations of motion, the relationship between ET and TAI can be expressed, to an accuracy that is sufficient for the purposes of the Pioneer project18, as

ET − TAI = (32.184 + 1.657 × 10 −3sinE ) seconds, (4.5 )
E = M + 0.01671 sin M, (4.6 ) M = 6.239996 + 1.99096871 × 10−7t, (4.7 )
and t is ET in seconds since the J2000 epoch (noon, January 1, 2000). For further details, including higher accuracy time conversion formulae, see the relevant literature [89909192235Jump To The Next Citation Point240Jump To The Next Citation Point] (in particular, see Equations (2–26) through (2–28) in [240Jump To The Next Citation Point].)

There exist alternate expressions with up to several hundred additional periodic terms, which provide greater accuracies. The use of these extended expressions provide transformations of ET − TAI to accuracies of 1 ns [240Jump To The Next Citation Point].

For the purposes of the investigation of the Pioneer anomaly, the Station Time (ST) is especially significant. The station time is the time kept by the ultrastable oscillators of DSN stations, and it is measured in Universal Coordinated Time (UTC). All data records generated by DSN stations are timestamped using ST, that is, UTC as measured by the station’s clock.

UTC is a discontinuous time scale; it is similar to TAI, except for the regular insertion of leap seconds, which are used to account for minute variations in the Earth’s rate of rotation. Converting from UTC to international atomic time (TAI) requires the addition or subtraction of the appropriate number of leap seconds (ranging between 10 and 32 during the lifetime of the Pioneer missions.) For more details see [240Jump To The Next Citation Point331].

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