Thursday, November 9, 2017

Measuring Planetary Spin from Spacecraft Timing

Exploring spacetime with general relativity

The trajectory of NASA's Juno Mission
General relativity is slowly becoming a tool that teaches us about the objects that curve spacetime instead of a hindrance to be corrected for. Planets are heavy and bend the fabric of spacetime affecting the orbits of satellites that go around them and the paths of light sent. Knowing very precisely when the light arrived, and when it was sent is part of space-craft timing. For Juno and Cassini this timing is already sensitive to higher order general relativistic effects like frame dragging. In the future, such measurements could be used to determine the spin of planets to within a percent, which can tell us about their interior. For missions in eccentric orbits relativistic effects are kick-like and can only be observed when the satellite is within a few hours of its pericentre.  We argue that instead of performing cumulative frame dragging measurements over many orbits as was done for Earth, high eccentricity missions like Juno and Cassini need the specific time dependence of each relativistic effect to aid in recovery.

A few words about gravity...

Bent spacetime affecting the orbit of a satellite & the light it sends
General relativity is the theory of gravitation. It says we live in four dimensions -- 3 spatial and 1 temporal -- and that space and time are connected. Gravitation itself is a consequence of the curvature of the spacetime. A planet is heavy and curves the spacetime around it causing it to seem like it pulls objects towards/around it.  This is gravity.

Redshift contribution from frame dragging for Juno. Zoom in.
Looking at relativistic effects
Missions like Juno and Cassini present new possibilities for measuring relativistic effects around the giant planets in our solar system. Relativistic effects are amplified if the orbit of the satellite is eccentric because the spacecraft moves faster and the satellite passes by the planet very closely where the gravitational field is stronger.  This coupled to the larger size of the planet causes frame dragging accelerations that are a few hundred times larger than those near Earth. Since the effects are larger, they might be easier to detect than around Earth.

Most planets have higher spins than black holes. The angular momentum per unit mass for black holes is less than 1, whereas for planets it can be of order hundreds. Earth has a spin on about 800 while Saturn's is about 1000. Precise frame dragging measurements can constrain planetary spin providing an independent estimate of the internal structure of the planet. This structure is relatively uncertain for the gas giants, which are believed to have an internal core of unknown size that rotates at a different rate than the surface.

 We simulate the trajectory of a satellite in a curved spacetime and find the path of the light it sends to receiving stations on Earth. Both follow 'straight lines' in their spacetime also called geodesics. The dynamics of a satellite orbiting a planet can largely be described by Newtonian physics with general relativity providing only small contributions.  The equations of motion are expanded in velocity orders to separate Newtonian and relativistic effects.

The biggest general relativistic effect is time dilation. GPS satellites are sensitive to time dilation and correct for it -- if they would not, the GPS would be off by about 10 km every day. Moving clocks tick slower than stationary clocks. So do clocks in a gravitational field. The ground station has its own time dilation and the difference between its tick signals and those arriving from the satellite are known as the redshift. We compute this redshift for satellites around Earth (Galileo and a proposed mission in an eccentric orbit) and for eccentric orbits around Jupiter (Juno) and Saturn (Cassini). Galileo satellites and the Atomic Clock Ensemble in space -- an ensemble of two atomic clocks that will be placed on the International space station in 2018 -- provide an even better measurements of time dilation, which tests the equivalence principle.

We are interested in higher order relativistic effects like frame dragging in which a spinning mass drags the spacetime in its vicinity affecting any orbiting satellite. The orbital plane of the satellite precesses about the spin axis of the planet. Historically, this effect was first predicted by in 1918 by Einstein, Lense and Thirring. They studied Amalthea, the third moon of Jupiter, and found that it precesses by 1'53'' per century.

Existent Measurements of frame dragging 
Orbital perturbations due to frame dragging have been measured using laser ranging by LARES and LAGEOS. Gravity Probe B measured the effects of frame dragging on the orientation of onboard gyroscopes. The effect is typically averaged over multiple orbits. It is then buried in much larger non-relativistic precession making it very hard to identify the relativistic contribution. E.g., Mercury's observed precession is mostly due to Newtonian planetary perturbations with the relativistic contribution being only about 7% of the total.

Relativistic effects for the Juno orbiter
Instead of averaging we compute each higher order relativistic effect as a function of time and find that they alter the orbit in a kick-like manner at the pericentre. For Juno the kick due to frame dragging could be measured for about two hours. We argue that technology has advanced enough so that we might be able to filter out these effects if we knew their specific time dependence.

This post summarizes results of:

Andreas Schaerer, Ruxandra Bondarescu, Prasenjit Saha, Raymond Angelil,  Ravit Helled and Philippe Jetzer, "Prospects for measuring Planetary Spin and Frame dragging in Spacecraft Timing Signals", Frontiers in Astronomy 4, 11 (2017).

Please read our article for more details.

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