Computation of Barycentric Dynamical Time using SPICE's definition

Question

I've written a high fidelity time management library (hifitime, currently in Rust, Python interface planned) which converts between time systems (TT, TAI, UTC, etc.) and time representations (Gregorian, JDE, MJD).

The current TDB computation uses ESA's Navipedia documentation. I found something similar in SPICE's documentation, but I have a hard time figuring out how to use that to convert between a TT epoch and a TDB epoch: from my validation examples, there's a very small difference between ESA and NASA's computation.

Could someone provide me an example of the execution of the SPICE algorithm if I have a TT epoch in seconds past J2000 and would like SPICE's TDB?

Answer 1

Neither one is correct. The two use slightly different approximations. SPICE's documentation, which you have found, uses $$TDB - TDT = K * sin (E)$$ where "$K$ is a constant and $E$ is the eccentric anomaly of the heliocentric orbit of the Earth-Moon barycenter". SPICE approximates the eccentric anomaly via $$E = M + e \sin (M)$$. This is incorrect. The correct expression is Kepler's equation, $$M = E - e\sin E$$. Inverting this to compute $E$ from $M$ (the mean anomaly) is the Kepler problem. The expression used by JPL is approximately correct for nearly circular orbits.

Navipedia uses a slightly different approximation, but it too is an approximation. It uses slightly different constants. Both approximations assume that TDB and TDT differ based on the distance and velocity between the Earth and the Sun, and that the Earth's orbit can be modeled as Keplerian. TDB and TDT instead differ based on the distance and velocity between the Earth and the solar system barycenter. Moreover, the Earth's orbit about the Sun is not quite Keplerian.

Answer 2

If you want high fidelity, you need to consult a more detailed reference, such as the International Astronomical Union's Standards Of Fundamental Astronomy. The SOFA Time Scale and Calendar Tools most recent revision is dated June 2020, and says, for example:

Of the seven time scales to be described here, one is atomic time (TAI), one is solar time (UT1), one is an atomic/solar hybrid (UTC) and four are dynamical times (TT, TCG, TCB, TDB). Each has a distinct role, and there are offsets of tens of seconds between some of them: when planning an astronomical calculation it is vital to choose the right one. A particularly common mistake is to assume that there is just one sort of precise time, namely UTC, compatible with everything from telescope pointing (which actually requires UT1) to looking up planetary positions (which requires TDB, which may be approximated by TT).

In fact, don't write the conversion equations yourself! Download the free C code for SOFA from https://www.iausofa.org/current.html , and just write a wrapper around it for your other languages.