# Determination of High-Accuracy Distances of Terrestrial Planets from the Sun

I understand how in olden days the Sun-Planet distances were estimated using:-

(i) measures of planet orbital periods ($$T$$) from analysis of observations over several centuries;

(ii) Kepler's 3rd Law $$(T^2 = k.a^3)$$ applied to the planet orbital periods to determine the relative lengths of the Sun-Planet semi-major axis distances ($$a$$) expressed in terms of the AU (astronomical unit = distance from centre of Earth to centre of Sun);

(iii) particular Earth-Planet distances along Solar radials were measured using Parallax techniques such as that by Casinni for Mars and Transits of Venus;

(iv) From one or more Earth-Planet radial distances the length of the AU could then be determined by simple algebra.

(v) Other Sun-Planet distances could then be determined once the AU length was found.

Modern ephemerides e.g. those issued by JPL use various techniques:-

The orbits of the inner planets are known to subkilometer accuracy through fitting radio tracking measurements of spacecraft in orbit about them. Very long baseline interferometry measurements of spacecraft at Mars allow the orientation of the ephemeris to be tied to the International Celestial Reference Frame with an accuracy of 0′′.0002. This orientation is the limiting error source for the orbits of the terrestrial planets, and corresponds to orbit uncertainties of a few hundred meters.

The orbits of Jupiter and Saturn are determined to accuracies of tens of kilometers as a result of fitting spacecraft tracking data.

The orbits of Uranus, Neptune, and Pluto are determined primarily from astrometric observations, for which measurement uncertainties due to the Earth’s atmosphere, combined with star catalog uncertainties, limit position accuracies to several thousand kilometers.

from Folkner et al 2014.

I understand how spaceraft telemetry has been used to obtain highly accurate distances between Earth and the terrestrial planets. But I am not clear about how highly-accurate distances can be determined between those planets and the centre of the Sun.

Question

Do such (Sun-planet-distance) determinations still basically rely on (the more accurate Newtonian version of) Kepler's 3rd Law relating planet orbital period and Sun-planet semi-major axis distance?

$$\frac{T^2}{a^3} = \frac{4 \pi^2}{G(M+m)}$$

What other methods or assumptions are used/involved?

EDIT - Afterthoughts

After further thought prompted by feedback from u/atmosphericprisonescape my question boils down to "How are the positions (over time) of the Sun's Centre tied-in (with sub-kilometre accuracy) to the 4D space-time array of triangulated, high-accuracy, telemetry-derived, Terrestrial planet position determinations.

I guess that Kepler's 3rd Law is not invoked as such. Rather, high-accuracy (sub-kilometre) determinations of the position of the Solar centre presumably require (in addition to the high-accuracy planet positions) some specific deterministic "motivation model"; i.e. a model of motion-determining factors.

These factors would include Newtonian Inertia, Newtonian Gravitational forces and Non-Newtonian factors leading to additional orbital angular velocity (cf Non-Newtonian Perihelion Precession and General Relativity as described at wikipedia/Apsidal_precession ).

The "motivation model" would constrain the relative positions of the Sun and planets via their involment as motion generators, reactors and enactors.

• "But I am not clear about how highly-accurate distances can be determined between those planets and the centre of the Sun. " Did you think about triangulation? – AtmosphericPrisonEscape Jan 5 '18 at 11:39
• @AtmosphericPrisonEscape. Yes I have thought about it and but I don't know what the levels of accuracy/error are for angular measurements and timing synchronization between measurements of different things (distance, angle) at different epochs. Also I'd guess there's some dependence on initial assumptions about observer position. Also afaik none of the raw triangulation data include direct measures of the position of the Solar centre. – steveOw Jan 5 '18 at 14:03
• We do live in the age where communication with light-speed is possible. Also we have satellites. Simultaneous observations of two objects on two different hemispheres is not a problem. Knowing this, I wouldn't think your issue to be a big problem, although I don't claim to know how your question can be answered. – AtmosphericPrisonEscape Jan 5 '18 at 14:16
• @AtmosphericPrisonEscape Thankx for your feedback. It has prompted me to add some Aftrthoughts to my question. :) – steveOw Jan 5 '18 at 17:20
• I might be missing something, but if we accurately know the Earth-Sun vector with accuracy e1 and the Earth-given_planet vector with accuracy e2, we know the Sun-given_planet vector with accuracy better than e1+e2 by the triangle inequality. – user21 Jan 6 '18 at 12:17

The DE430 JPL ephemeris "memo" (https://ipnpr.jpl.nasa.gov/progress_report/42-196/196C.pdf) has details of how it is constructed and what from. Mercury and Venus's positions are tracked by tracking the spacecraft orbiting them (to sub-km accuracy). Mars, Jupiter and Saturn are also measured from the orbiting spacecraft such as Galileo and Cassini. The outer planets plus Pluto positions are primarily done from astrometry so measuring the positions of the planets relative to the background stars, in the past through photographic plates or with CCDs in more modern times.

Positions of stars and planets are all measured in something called the International Celestial Reference System (ICRS) which is based on measuring very distant and fixed on the sky objects called quasars using a precise radio technique called VLBI.

• Thanks - that memo is the one I linked as "Folkner et al 2014". It seems to imply that the accurate estimated position of the Sun (relative to Earth, for example) is dependent on the "dynamical model" which they use. (More or less what I was hypothesizing - as the "motivational model" - in the afterthoughts section of my question.) Thus it seems to me that high-resolution (sub-Km) estimates of Sun-to-Planet distances are sensitive to the detailed gravitation model used (while Planet-Planet distances are not). – steveOw Jul 25 '18 at 13:24
• Checking Chap. 8 of Ref [4] in Folkner et al. 2014 goes into some 40 pages of details. The observations are fit through parametrized post-Newtonian (PPN) n-body equations for each body (planets, Moon, big asteroids). Big G (the gravitational constant) is included and PPN has 2 other parameters that let you do non-Newtonian gravity. They state that both are set to 1 as in GR. There is a separate equation for the barycenter and the position and velocity of the Sun are obtained from this and not integrated. So positions, separations etc are all fitted to the data via assumed model and constants – astrosnapper Jul 25 '18 at 17:06
• Thanks for pointing that section in the Standish-Williams, 2012 reference. Very helpful (although I am not clear - from their brief description - about all the details). Interesting to see that the barycentre determination involves relativistic (velocity-dependent) factors. (Answer Accepted.) – steveOw Jul 25 '18 at 17:59