# How long would one orbit around our galaxy's supermassive black hole be in the reference frame of S2

Physics Asked by Mape on October 16, 2020

With the 2020 Nobel Prize in physics being awarded for the proof and discovery of supermassive black holes, I started doing some research and came across the star closest to our own galaxy’s supermassive black hole, S2.

S2 seen from Earth makes one orbit in 16.0518 years. How long time would one orbit take in the reference system of the star?

This made me think of the movie ‘Interstellar’ where one hour spent on Miller’s planet close to the black hole Gargantua equaled 7 years on Earth. How does this star come in comparison?

The S2 star is still very far away from the black hole in terms of gravitational radii. A quick back of the envelope estimate gives that the proper time period of S2 is about 8 hours short than its 16 year orbital period.

Update: The comments asked for more details. The solutions for (test particle) orbits around a black hole are known in closed form, including expressions for the period measured in terms of the proper time of the particle and the time of an asymptotic observer. I'm not going to repeat the expressions here, but a complete collection can be found in my paper from 2019. They are also coded up in the Black Hole Perturbation Toolkit for general use.

Using these and estimates for the system parameters (Black hole mass $$sim4.25cdot10^6 M_{odot}$$, eccentrity 0.88466, and pericenter distance of 120 au) one can calculate the difference between the two periods, which comes in at about 8 hours.

The main source of uncertainty here are the various source parameters. Furthermore, we neglect the influence of other objects on the orbit.

Correct answer by mmeent on October 16, 2020

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