[Paper Review] Some aspects on the observation of the gravitomagnetic clock effect

Herbert Lichtenegger, W. Hausleitner+2 more|ArXiv.org|Jan 23, 2001

Geophysics and Sensor Technology11 references3 citations

TL;DR

This paper proposes using orbiting clocks to detect Earth's gravitomagnetic field via the gravitomagnetic clock effect, where co- and counter-orbiting satellites exhibit a measurable period difference due to Earth's rotation. The key finding is that mismodeling of Earth's gravitational field—especially even zonal harmonics $C_{n0}$ and sectoral terms $C_{n2}, S_{n2}$—introduces errors up to 1000 times larger than the target effect, making precise gravity field knowledge essential for detection.

ABSTRACT

As a consequence of gravitomagnetism, which is a fundamental weak-field prediction of general relativity and ubiquitous in gravitational phenomena, clocks show a difference in their proper periods when moving along identical orbits in opposite directions about a spinning mass. This time shift is induced by the rotation of the source and may be used to verify the existence of the terrestrial gravitomagnetic field by means of orbiting clocks. A possible mission scenario is outlined with emphasis given to some of the major difficulties which inevitably arise in connection with such a venture.

Motivation & Objective

• To assess the feasibility of observing the gravitomagnetic clock effect using counter-orbiting atomic clocks in Earth orbit.
• To quantify how inaccuracies in Earth's gravitational field model—particularly $C_{n0}$, $C_{n2}$, and $S_{n2}$—affect the detection of the clock effect.
• To determine whether secular and periodic perturbations in orbital periods can be modeled well enough to isolate the gravitomagnetic signal.
• To evaluate the required accuracy in gravity field coefficients for a successful experiment, given the small magnitude of the clock effect (~2×10⁻⁷ s).

Proposed method

• Modeling the orbital period perturbations due to Earth's static gravitational field using the EGM96 gravity model up to degree and order 6.
• Calculating secular period shifts via the perturbation term $\Delta P_{nm}^{sec} \propto C_{nm} \cdot R_E^n / a^n$ for zonal ($m=0$) and sectoral ($m=n$) harmonics.
• Estimating maximum periodic perturbations using the amplitude $|\Delta P_{nm}^{max}| = (C_{nm}^2 + S_{nm}^2)^{1/2}$, with periods derived from orbital dynamics.
• Assessing mismodeled period errors by comparing predicted shifts to the expected gravitomagnetic clock effect of $\sim 2 \times 10^{-7}$ s.
• Evaluating the impact of orbital radius (7000, 8000, 12000 km) on error magnitude, showing higher altitudes reduce sensitivity to high-degree harmonics.
• Using the PPN formalism to confirm that the clock effect depends on the same parameter combination as Lense-Thirring precession, validating its relativistic origin.

Experimental results

Research questions

• RQ1 Can the gravitomagnetic clock effect be detected using counter-orbiting atomic clocks in Earth orbit, given current gravity field models?
• RQ2 How do uncertainties in the Earth's spherical harmonic coefficients $C_{n0}$, $C_{n2}$, and $S_{n2}$ affect the accuracy of period measurements needed to detect the clock effect?
• RQ3 To what extent do secular and periodic gravitational perturbations in orbital periods mask or distort the gravitomagnetic signal?
• RQ4 What level of improvement in Earth's gravitational field model is required to reduce mismodeling errors below the threshold of the clock effect?
• RQ5 Does the accumulative nature of the clock effect over multiple revolutions help mitigate periodic perturbations, and if so, under what conditions?

Key findings

• The gravitomagnetic clock effect produces a period difference of approximately $2 \times 10^{-7}$ s for Earth-orbiting satellites, independent of orbital radius.
• Secular perturbations due to $C_{20}$ cause a period shift of $-15.7$ s for a 7000 km orbit, which is over 100 million times larger than the target signal.
• Mismodeling of $C_{n0}$ coefficients up to degree 100 leads to period errors exceeding the required accuracy for detection in low Earth orbits (e.g., 7000 km).
• Uncertainties in $C_{n2}$ and $S_{n2}$ coefficients produce periodic perturbations with maximum amplitude errors up to 1000 times larger than the clock effect at 7000 km altitude.
• At 12,000 km altitude, the required accuracy for $C_{n0}$ coefficients is reduced to degree ~20, and mismodeling errors from $C_{n2}, S_{n2}$ drop to ~100 times the signal level.
• Despite large periodic mismodeling, these effects may cancel out over multiple revolutions due to the accumulative nature of the clock effect, but secular errors remain a dominant obstacle.