---
title: Areocentric Coordinate Time (TCA) on Mars
url: https://www.emergentmind.com/topics/areocentric-coordinate-time-tca
type: topic
---

# Areocentric Coordinate Time (TCA) on Mars

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Searching arXiv by title and topic keywords: "High-Precision Relativistic Time Scales for Mars Surface and Orbital Clocks", "Relativistic Time Scales and Transformations in the Solar System", and Mars coordinate time.
Areocentric Coordinate Time (TCA) is a Mars-centered relativistic coordinate time associated with the Mars-centered celestial reference system (MCRS). It is introduced as the Mars analogue of Geocentric Coordinate Time (TCG), while related recent work denotes the corresponding Mars body-centered coordinate time as MCG within the Mars Celestial Reference System. In both formulations, TCA is a precision reference for time transfer, navigation, ephemeris reduction, and radiometric tracking around Mars; it is explicitly not Mars Solar Time, Mars Sol Date, or a coordinated civil Mars time, and it is not a universal “master clock” [2606.13726; 2607.00550].

## 1. Concept and nomenclature

TCA belongs to a relativistic hierarchy in which each observable is characterized by proper time on a world line, coordinate time in a relevant celestial reference system, and the transformations between them. For Mars, recent work places the Mars-centered coordinate time between local clock proper time and the solar-system barycentric time used by ephemerides and tracking reductions. In the notation of one formulation, the chain is
\[
\tau \rightarrow \mathrm{MCG} \rightarrow \mathrm{TCB} \rightarrow \mathrm{TT/UTC},
\]
while another formulation distinguishes \(TCA\), a conventional Mars surface scale \(T_M\), and clock proper time \(\tau\) [2607.00550; 2606.13726].

This structure is central to the interpretation of TCA. Proper time is what a physical clock measures along its world line; TCA is the coordinate time assigned by a Mars-centered relativistic reference system. A plausible implication is that TCA should be understood primarily as an element of reference-system architecture rather than as a directly observable clock reading.

Two misconceptions are explicitly excluded in the recent literature. First, TCA is not a civil or solar timescale for day-to-day scheduling on Mars. Second, Mars timekeeping cannot be reduced to a single universal clock. The relevant consistency condition is instead a documented transformation chain linking a specified world line, a body-centered coordinate time, the barycentric frame, and operational atomic scales at tracking stations [2607.00550].

## 2. Mars-centered reference systems and relativistic definition

The modern definition of TCA specializes the International Astronomical Union BCRS/GCRS post-Newtonian formalism to Mars. In the barycentric frame, the coordinate time is \(t \equiv TCB\). In the Mars-centered system, the coordinate time is
\[
T \equiv TCA.
\]
The MCRS is taken to be kinematically non-rotating relative to the BCRS, while a body-fixed Mars frame is obtained by an IAU/WGCCRE rotation model. Proper-time calculations are stated to be cleanest in the MCRS, whereas operational realizations must also use a consistent ephemeris, Mars orientation model, gravity model, and spacecraft orbit-determination solution [2606.13726].

The BCRS metric is written in standard IAU form as
\[
g_{00}=1-\frac{2w}{c^2}+\frac{2w^2}{c^4}+(c^{-5}),\qquad
g_{0i}=-\frac{4w^i}{c^3}+(c^{-5}),\qquad
g_{ij}=-\delta_{ij}\left(1+\frac{2w}{c^2}\right)+(c^{-4}),
\]
and the Mars-centered metric is written analogously as
\[
G_{00}=1-\frac{2}{c^2}\Big(U_{Ma}+U_{\rm tid}\Big) +\frac{2U_{Ma}^2}{c^4}+(c^{-5}),
\]
\[
G_{0i}=-\frac{4}{c^3}W^i_{Ma}+(c^{-5}),\qquad
G_{ij}=-\delta_{ij}\Big\{1+\frac{2}{c^2}\Big(U_{Ma}+U_{\rm tid}\Big)\Big\}+(c^{-4}).
\]

The corresponding Mars-centered time transformation is the Mars analogue of the IAU Earth-centered transformation:
\[
T = t - \frac{1}{c^2}\left[A(t)+v_{Ma}^i r_{Ma}^i\right]
+\frac{1}{c^4}\left[B(t)+B^i(t)r_{Ma}^i+B^{ij}(t)r_{Ma}^i r_{Ma}^j+C(t,\mathbf{x})\right] +(c^{-5}),
\]
with \(r_{Ma}^i=x^i-x_{Ma}^i(t)\), and with the coefficients constrained by
\[
\frac{d A}{d t} = \frac{1}{2}v_{Ma}^2+\bar w_{\rm ext}(\mathbf{x}_{Ma}),
\]
\[
\frac{d B}{d t} = -\frac{1}{8}v_{Ma}^4 -\frac{3}{2}v_{Ma}^2\bar w_{\rm ext}(\mathbf{x}_{Ma}) +4v_{Ma}^i\bar w^i_{\rm ext}(\mathbf{x}_{Ma}) +\frac{1}{2}\bar w_{\rm ext}^2(\mathbf{x}_{Ma}).
\]
Accordingly, TCA is not a constant offset from TCB; it includes secular barycentric terms, periodic orbital terms, the event-dependent term \((\mathbf{v}_{Ma}\cdot\mathbf{r}_{Ma})/c^2\), formally retained \(c^{-4}\) contributions, and higher-order remainders [2606.13726].

## 3. Transformation hierarchy: TCB, TCA, \(T_M\), and proper time

The Mars timing framework distinguishes three layers. The first is TCA itself, the coordinate time of the MCRS. The second is a conventional Mars surface scale \(T_M\), introduced in explicit analogy with terrestrial constant-rate rescalings. The third is the proper time \(\tau\) of a realized landed or orbiting clock [2606.13726].

The conventional surface scale is defined by
\[
\Big\langle\frac{d T_M}{d TCA}\Big\rangle=1-L_{\rm surf},
\]
with the adopted provisional conventional value
\[
L_{\rm surf}^{\rm def}=1.406355\times10^{-10} =12.15091\,\mu{\rm s\,d^{-1}}.
\]
The associated scaling is
\[
T_M = TCA - L_{\rm surf}^{\rm def}(TCA-T_{M0}), \qquad
\mathbf{X}_M = (1-L_{\rm surf}^{\rm def})\,\mathbf{X}_{TCA}, \qquad
(GM)_M = (1-L_{\rm surf}^{\rm def})(GM)_{TCA}.
\]

A landed clock does not, in general, realize \(T_M\) exactly. Its proper-time rate relative to the adopted surface scale depends on the local potential departure from the reference areoid:
\[
\frac{d\tau_{\rm lander}}{d T_M}-1 = -\frac{\Delta W_{\rm lander}}{c^2} + (c^{-4}),
\qquad
\Delta W_{\rm lander}=W_{\rm lander}-W_0^{Ma}.
\]
For orbiting clocks, the Mars-centered proper-time law is
\[
\frac{d\tau}{d T}=1-\frac{1}{c^2}\Big\{\frac{1}{2}V^2+U_{Ma}(T,\mathbf{X})+U_{\rm tid}(T,\mathbf{X})\Big\}+(c^{-4}).
\]

A parallel formulation expresses the Mars body-centered coordinate time as a provisional MCG tied to the MCRS and obtained from TCB by integrating the relevant 1PN rate along the areocentre world line in the BCRS, using DE440-class ephemeris integrations. This reinforces the same point: the Mars-centered timescale is defined by a body-centered relativistic transformation chain, not by a standalone clock convention [2607.00550].

## 4. Secular and periodic clock-rate structure

A defining feature of recent Mars timing work is the decomposition of clock transformations into secular rates plus zero-mean periodic terms. In the compact notation used for the Mars-centered program,
\[
\tau-T_{\rm ref}=\Delta_{\rm sec}(t)-P_{\rm common}(t)-P_{\rm orb}(t)-\Delta_{\rm geom}(t)+\epsilon(t),
\]
and for the TCA–TCB link the Mars barycentric energy function satisfies
\[
\mathcal{E}_{Ma}(t)=L_A+\dot P_A(t), \qquad \langle \dot P_A\rangle=0.
\]
The associated “Mars time ephemeris” construction integrates
\[
\mathcal{E}_{Ma}(t) =\frac{1}{c^2}\left\{\frac{1}{2}v_{Ma}^2+\bar w_{\rm ext}(\mathbf{x}_{Ma})\right\}
+\frac{1}{c^4}\left\{\frac{1}{8}v_{Ma}^4 +\frac{3}{2}v_{Ma}^2\bar w_{\rm ext}(\mathbf{x}_{Ma}) -4v_{Ma}^i\bar w^i_{\rm ext}(\mathbf{x}_{Ma}) -\frac{1}{2}\bar w_{\rm ext}^2(\mathbf{x}_{Ma})\right\},
\]
defines
\[
I_A(t)=\int_{t_0}^{t}\mathcal{E}_{Ma}(t')\,dt',
\]
and splits the result as
\[
I_A(t)=L_A(t-t_0)+P_A(t)-P_A(t_0).
\]
The Sun-only secular estimate is
\[
L_A^{(\odot)}\simeq\frac{3GM_\odot}{2a_{Ma}c^2}=9.72\times10^{-9} =0.8396\,\mathrm{ms\,d^{-1}},
\]
with leading eccentricity modulation
\[
\delta \dot P_A \simeq \frac{2GM_\odot}{a_{Ma}c^2}e_{Ma}\cos M_{Ma},
\]
having amplitude about \(1.21\times10^{-9}\), or \(104.6\,\mu\mathrm{s\,d^{-1}}\), and annual timing amplitude about \(11\,\mathrm{ms}\) [2606.13726].

The Mars surface rate relative to a geoid-referenced terrestrial clock is also quantified directly. Using a surface potential approximation together with the centrifugal correction, one estimate gives
\[
\frac{d\tau_{\mathrm{M}}}{d\tau_{\oplus}} \approx 1 + \frac{\Delta w_{\mathrm{M}\oplus}}{c^{2}}
\approx 1 + 5.6\times 10^{-10},
\]
with the more precise monopole estimate
\[
\Delta w_{\mathrm{M}\oplus}/c^{2} = 5.558\times 10^{-10},
\]
corresponding to
\[
48.0\,\mu\mathrm{s}\,\mathrm{day}^{-1}.
\]
The Mars centrifugal correction is stated to be much smaller, about \(3.2\times10^{-13}\) in \(d\tau/dt_{\mathrm{MCG}}-1\) [2607.00550].

## 5. Gravity-field realization, tides, and retained terms

The Mars-centered realization of TCA is explicitly model-based. One high-precision construction retains terms when their fractional-frequency amplitude exceeds \(5\times10^{-18}\) or their one-way accumulated timing amplitude exceeds \(0.1\) ps:
\[
\epsilon_f=5\times10^{-18},\qquad \epsilon_t=0.1\,\mathrm{ps}.
\]
The same work notes that \(5\times10^{-18}\) per day corresponds to \(0.432\) ps/day, and that \(0.1\) ps corresponds to \(0.030\) mm one-way light-travel distance [2606.13726].

The numerical realization uses the GMM-3 Mars gravity field through degree and order 120, point-mass tides from the Sun, Phobos, and Deimos with origin and dipole terms removed, and bounds on omitted local \(c^{-4}\) and external-perturber terms. The static Mars potential is represented as
\[
U_{Ma}(r,\theta,\lambda)=\frac{GM_{Ma}}{r}\left[1+\sum_{\ell=2}^{\ell_{\max}}\sum_{m=0}^{\ell}\left(\frac{R_0}{r}\right)^\ell \bar P_{\ell m}(\cos\theta)\left(\bar C_{\ell m}\cos m\lambda+\bar S_{\ell m}\sin m\lambda\right)\right],
\]
with the degree-2 coefficient
\[
J_{2Ma}\simeq1.9566\times10^{-3}.
\]
The residual-potential gate is
\[
\frac{|\Delta U|}{c^2}\le 5\times10^{-18}, \qquad |\Delta U|\le0.45\,\mathrm{m^2\,s^{-2}}.
\]

Time-variable low-degree gravity is a prominent realization issue. Seasonal CO\(_2\) exchange and atmospheric loading are identified as leading terms for a realized Mars surface scale, capable of producing fractional-rate shifts of \(10^{-18}\)–\(10^{-17}\). The formalism therefore introduces seasonal and tidal harmonic variations through
\[
\bar C_{\ell m}(T)=\bar C_{\ell m}^{(0)} +\Delta\bar C_{\ell m}^{\rm seas}(T) +\Delta\bar C_{\ell m}^{\rm tide}(T),
\]
\[
\bar S_{\ell m}(T)=\bar S_{\ell m}^{(0)} +\Delta\bar S_{\ell m}^{\rm seas}(T) +\Delta\bar S_{\ell m}^{\rm tide}(T).
\]

The external tidal potential is written as
\[
U_{\rm tid}(T,\mathbf{X})=\sum_{B\ne Ma}\sum_{\ell=2}^{N_B}\frac{GM_B}{r_{BMa}}\left(\frac{X}{r_{BMa}}\right)^\ell P_\ell(\hat{\mathbf{n}}_{BMa}\cdot\hat{\mathbf{X}}),
\]
and the solar quadrupole tide at mean Mars distance is about \(1.4\times10^{-18}\) in fractional rate, rising to \(\sim1.9\times10^{-18}\) at perihelion through
\[
f_{\rm peri}=(1-e_{Ma})^{-3}\simeq1.342.
\]
At mean distance, the same tide scales with \(X^2\), becoming \(5.2\times10^{-17}\) at areostationary radius and \(6.9\times10^{-17}\) at Deimos distance; the corresponding perihelion values are \(7.0\times10^{-17}\) and \(9.3\times10^{-17}\). Jupiter’s local tide is stated to be \(<3.6\times10^{-21}\) for the simulated regimes and is therefore neglected locally [2606.13726].

## 6. Representative orbital regimes and operational consequences

Recent Mars timing studies evaluate representative surface and orbital regimes relative to the adopted Mars surface scale \(T_M\). The resulting offsets are large on the scale of high-precision time transfer and are accompanied by retained periodic structure [2606.13726].

| Regime | Mean offset relative to \(T_M\) | Selected retained or notable term |
|---|---:|---|
| 300 km low Mars orbit | slower by \(4.56\,\mu\mathrm{s\,d^{-1}}\) | Mars \(J_2\) line \(\simeq 87\) ps |
| Areostationary orbit | faster by \(9.13\,\mu\mathrm{s\,d^{-1}}\) | solar quadrupole \(0.28\) ps mean, \(0.38\) ps perihelion |
| Phobos-distance orbit | faster by \(5.56\,\mu\mathrm{s\,d^{-1}}\) | \(J_2\) line \(\sim 21.5\) ps |
| Deimos-distance orbit | faster by \(9.52\,\mu\mathrm{s\,d^{-1}}\) | solar tide \(\sim 0.45\) ps mean, \(0.60\) ps perihelion |
| Highly elliptical relay orbit | faster by \(7.02\,\mu\mathrm{s\,d^{-1}}\) | Keplerian excursion \(\simeq 0.350\,\mu\mathrm{s}\) |

The highly elliptical relay case is especially diagnostic. For the orbit with periareion \(300\) km, apoareion \(17000\) km, \(a=12046.2\) km, \(e=0.6932\), and period \(11.15\) h, the first Fourier amplitudes are
\[
A_1\simeq0.330\,\mu\mathrm{s},\qquad
A_2\simeq0.103\,\mu\mathrm{s},\qquad
A_3\simeq0.0478\,\mu\mathrm{s},
\]
with one-way range equivalents stated to be tens to hundreds of meters [2606.13726].

These timing structures matter operationally because Mars tracking reductions are fundamentally barycentric. The Mars-centered coordinate label is therefore only one part of a longer chain involving TCB, TT/UTC, link modeling, and proper time on spacecraft and station world lines. The null-geodesic light-time relation, Shapiro delay, and two-way range-rate expressions in the Mars radiometric model lead to Mars-range Shapiro-rate terms at the \(10^{-12}\) to \(10^{-13}\) level, and inconsistent time tagging can produce microsecond-level range biases, microsecond-level Doppler biases, and corrupted inter-agency or multi-mission data fusion [2607.00550].

The broader conceptual template derives from body-centered relativistic timing work beyond Mars. The lunar framework shows how a body-centered coordinate time can be constructed by selecting a local inertial frame, defining a reference equipotential surface, compensating gravitational and rotational time dilation, and synchronizing a clock network to realize the resulting time scale. For Mars, however, the extension is explicitly described as more barycentric and more ephemeris-driven, because the Earth–Mars comparison is dominated by the Sun’s gravitational potential and does not benefit from the Earth–Moon system’s simpler local two-body symmetry [2402.11150].

Source: https://www.emergentmind.com/topics/areocentric-coordinate-time-tca