User:NOrbeck/Time ephemerides

Time ephemerides. Relativistic time dilation integral.

Introduction
Crucial for pulsar timing. Used to reduce observed Time of Arrival (ToA) to the solar system barycenter. For example, the Tempo2 software package.

The relativistic time dilation integral:


 * $$ TCB - TCG =

\frac{1}{c^2} \int_{t_0}^{t} \left( U_{ext}( \vec{x_\oplus} ) + \frac{ v_\oplus^2 }{2} \right) \operatorname{d}t +\frac{ \vec{r_\oplus} \vec{v_\oplus} }{c^{2}}  + \mathcal{O}(c^{-4}) $$
 * $$ T_\odot - T_\oplus =

\frac{1}{c^2} \int_{t_0}^{t} \left( U_{ext}( \vec{x_\oplus} ) + \frac{ v_\oplus^2 }{2} + \Delta L_c \right) \operatorname{d}t +  \mathcal{O}(c^{-4}) $$ where:
 * $$ t, t_0 $$ are the time arguments of the spatial ephemeris (Teph),
 * $$ T_\odot $$ is the barycentric coordinate time (TCB),
 * $$ T_\oplus $$ is the terrestrial coordinate Time (TT),
 * $$ \vec{x_\oplus} $$ is the barycentric position of the Earth,
 * $$ v_\oplus $$ is the barycentric velocity of the Earth,
 * $$ U_\oplus $$ is the Newtonian potential of all solar system bodies, excluding for the Earth, evaluated at the geocenter.
 * $$ \Delta L_c $$ is the time ephemeris correction constant,
 * $$ c $$ is the speed of light.

Relativistic scale factors

 * $$ L_c $$ = $1.481$ ± $2$ = the average value of 1 - d(TCG)/d(TCB)


 * $$ L_b $$ = $1.551$, mean rate of TCB-TT.
 * $$ L_b = 1 - (1-L_c)(1-L_g) = L_c + L_g - L_c L_g = 1 - \frac{ \operatorname{d} \mathrm{TDB} }{ \operatorname{d} \mathrm{TCB} } $$


 * $$ L_g $$ = $6.969$ (defining constant). IERS Conventions (2010)
 * $$ L_g = 1 - \frac{ \operatorname{d} \mathrm{TT}}{ \operatorname{d} \mathrm{TCG} } $$

TE405
TE405 is a numerical time ephemeris of the Earth based on the JPL DE405 ephemeris.

TE405 approximates the relativistic time-dilation integral from the years 1600 to 2200 using numerical quadrature of the DE405 ephemeris. The integral is required to transform between Terrestrial Time (TT), and the (solar system) barycentric time scales ephemeris time (Teph) or TCB. Teph is a linear transformation of TCB that represents the independent variable of a modern numerical ephemeris. The time ephemeris results have an accuracy of order 0.1 ns. Available here as a discrete Chebyshev approximation that requires much less computer time to evaluate than a detailed time ephemeris series. Angular frequency and mass transformation corrections that should be applied to the time ephemeris series of Fairhead & Bretagnon. These corrections make an extended form of this series with 1705 terms agree with our work to within 15 ns over the epoch range. A further correction of two long-term sinusoids that reduces this maximum residual to 5 ns. The long-term residuals fit by these sinusoids and the remaining short-term residuals appear to be the result of errors in the fit of VSOP82/ELP2000 (the analytical ephemeris upon which the Fairhead & Bretagnon series is based) to the earlier JPL ephemeris, DE200. Following Fukushima we eliminate the linear term from TE405 by comparing with the corrected series results. The result for the linear coefficient of the subtracted term is
 * $$ \Delta L_C(TE405) = 1.48082685594 \times 10^{-8} \pm 1 \times 10^{-17} $$.

This does not include the periodic post-Newtonian and asteroid perturbation effects because they are negligible. However, when the average post-Newtonian and asteroid corrections of Lc(PN) = $109.7$ and Lc(A) = $5$ to Lc, the result is Lc = $1.481$.

When this result is combined with a recent value for the potential at the geoid corresponding to Lg = $6.969$ ± $6$ one obtains,


 * $$ K = \frac{ \operatorname{d} TCB }{ \operatorname{d} T_\mathrm{eph} } = \frac{ \operatorname{d} TCB }{ \operatorname{d} ET } = \frac{1}{ (1-L_c)(1-L_g) } = 1 + (1.55051979154 \times 10^{-8} \pm 3 \times 10^{-17} ) $$.

The factor K relates ephemeris units for time and distance to the corresponding SI units for the same quantities.