Light-Time Model

Optical and radar observation models both need a light-time solution. The target state is not always evaluated at the reported observation epoch. It is evaluated at the signal emission or bounce epoch required by the signal path.

Optical One-Way Light Time

Optical observations use a one-way light-time solution. The observation epoch is the receive epoch \(t_{\mathrm{r}}\). The observer is evaluated at \(t_{\mathrm{r}}\), and the target is evaluated at the earlier emission epoch \(t_{\mathrm{r}} - \tau_{\mathrm{opt}}\). This gives the target direction seen by the observer.

DiffOrb solves the one-way light time from:

\[ \tau_{\mathrm{opt}} = {1 \over c} \left\| \boldsymbol{r}_{\mathrm{body}}(t_{\mathrm{r}} - \tau_{\mathrm{opt}}) - \boldsymbol{r}_{\mathrm{obs}}(t_{\mathrm{r}}) \right\| + \Delta\tau_{\mathrm{rel}} \]

Here \(c\) is the speed of light. \(\boldsymbol{r}_{\mathrm{body}}\) is the target position, and \(\boldsymbol{r}_{\mathrm{obs}}\) is the observer position. The term \(\Delta\tau_{\mathrm{rel}}\) is the Shapiro delay from gravitating bodies.¹

The optical one-way solution uses the dense target trajectory, the observer state at the receive epoch, and the time-scale rules that connect the observation time to TDB.

Radar Two-Way Light Time

Radar observations use a two-way light-time solution. The reference epoch is the receive epoch \(t_{\mathrm{r}}\). The signal path has a down leg from the target to the receiver and an up leg from the transmitter to the target.²

DiffOrb first solves the down-leg delay:

\[ \tau_{\mathrm{d}} = {1 \over c} \left\| \boldsymbol{r}_{\mathrm{body}}(t_{\mathrm{r}} - \tau_{\mathrm{d}}) - \boldsymbol{r}_{\mathrm{rx}}(t_{\mathrm{r}}) \right\| + \Delta\tau_{\mathrm{d,rel}} + \Delta\tau_{\mathrm{d,cor}} + \Delta\tau_{\mathrm{d,tropo}} \]

This defines the target bounce epoch:

\[ t_{\mathrm{b}} = t_{\mathrm{r}} - \tau_{\mathrm{d}} \]

DiffOrb then solves the up-leg delay:

\[ \tau_{\mathrm{u}} = {1 \over c} \left\| \boldsymbol{r}_{\mathrm{body}}(t_{\mathrm{b}}) - \boldsymbol{r}_{\mathrm{tx}}(t_{\mathrm{b}} - \tau_{\mathrm{u}}) \right\| + \Delta\tau_{\mathrm{u,rel}} + \Delta\tau_{\mathrm{u,cor}} + \Delta\tau_{\mathrm{u,tropo}} \]

The transmitter and receiver may be the same site for monostatic radar or different sites for bistatic radar. The radar delay is the solved round-trip light time.

The delay correction terms are applied separately to the down leg and the up leg. Relativistic delay uses the Shapiro model.¹ Solar-corona delay follows the standard solar-corona electron-density correction.³⁴ Tropospheric delay is applied near Earth stations.⁵

The radar two-way solution uses the dense target trajectory, transmitter and receiver site states, and the time-scale rules that connect TDB, TT, UTC, and UT1.

Radar Doppler Reduction

Radar Doppler reduction uses the same converged two-way light-time model as radar delay. Traditional formulations often write Doppler from analytic expressions for the rates of change of the up-leg and down-leg path lengths.²

DiffOrb instead treats the Doppler shift as the derivative of the converged round-trip delay with respect to the receive epoch. This derivative is computed by automatic differentiation. This keeps radar delay and radar Doppler tied to the same signal-path model.

Read Next

• Read Numerical Integrators And Dense Trajectories for the target trajectory queried by light-time iteration.
• Read Observer Site Keys And Observer Types for ground, roving, and space observer states.
• Read Ephemeris Products for the product correction levels built on this model.
• Use Get Radar Outputs In Monostatic And Bistatic Geometry for a concrete radar prediction path.

References

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1. Shapiro, I. I. (1964). Fourth Test of General Relativity. Physical Review Letters, 13(26), 789-791. https://doi.org/10.1103/PhysRevLett.13.789 ↩↩

2. Yeomans, D. K., Campbell, D. B., Chodas, P. W., Giorgini, J. D., & Ostro, S. J. (1992). Asteroid and Comet Orbits Using Radar Data. The Astronomical Journal, 103(1), 303-317. ↩↩

3. Muhleman, D. O., & Anderson, J. D. (1981). Solar wind electron densities from Viking dual-frequency radio measurements. The Astrophysical Journal, 247, 1093-1101. NASA NTRS record: https://ntrs.nasa.gov/citations/19810061604 ↩

4. Standish, E. M., & Williams, J. G. Orbital Ephemerides of the Sun, Moon, and Planets, in Explanatory Supplement to the Astronomical Almanac, especially Section 8.7.6. ↩

5. Urban, S. E., & Seidelmann, P. K. (eds.). Explanatory Supplement to the Astronomical Almanac, especially Section 8.7.5 on relativity and Section 8.7.7 on tropospheric delay. ↩