# Estimating Mass of Central Object Using S2's Orbit Motion and Kepler's 3rd Law

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• doopa
In summary, the conversation revolved around using Kepler's 3rd law in solar units to find the mass of a central object based on the orbit motion of star S2. The participants were given a hint to use the arcsec relation and read the radius of the orbit from the graph, with a suggested estimate of 0.1" for the semi-major axis. There was also a discussion about the possible tilt of the orbit and whether it needed to be corrected for in order to accurately estimate the mass of the central object.
doopa
TL;DR Summary
I want to understand how one can go about estimating the mass of a central object given the a graph the orbital motion of S2, or any given star for that matter.
During one of my class lectures, we were shown a graph of star S2's orbit motion (shown at the bottom of this post) and was told to try and figure out how to find the mass of the central object using Kepler’s 3rd law in solar units, as shown below:

We were also given a hint to use the arcsec relation and read the radius of the orbit from the image, as shown below:

From my notes, it looks like we had to use the declination value of 0.1, but I still don't understand how exactly we got to that point. Does anyone happen to know why this is and how to generally use these types of graphs to estimate the mass of a central object?

The "radius" in the formula is actually half the semi-major axis of the ellipse (which obviously doesn't have a constant radius). Based on the diagram, 0.1'' might be a good estimate for the semi-major axis.

As @pasmith said, reading from the diagram, the semimajor axis is about 0.1". This is the θ in your second formula. It is not a declination, it is the angular size of the orbit.

## 1. What is the "orbit" of S2?

The "orbit" of S2 refers to the path that the star S2 takes around the supermassive black hole at the center of our Milky Way galaxy. This orbit is highly elliptical, meaning that it is not a perfect circle, and takes approximately 16 years to complete.

## 2. How was the orbit of S2 interpreted?

The orbit of S2 was interpreted using data from the Very Large Telescope (VLT) in Chile, which has been tracking the star's movements since 1992. By measuring the star's position and velocity over time, scientists were able to plot its orbit and make predictions about its future movements.

## 3. What does the orbit of S2 tell us about the black hole at the center of our galaxy?

The orbit of S2 provides valuable information about the mass and gravitational pull of the black hole at the center of our galaxy. By studying the star's movements, scientists were able to calculate the mass of the black hole to be approximately 4 million times that of our Sun.

## 4. How does the orbit of S2 support the theory of general relativity?

The orbit of S2 is consistent with the predictions of Einstein's theory of general relativity, which describes how massive objects like black holes warp the fabric of spacetime. The star's orbit shows that it is affected by the strong gravitational pull of the black hole, just as predicted by this theory.

## 5. What are the potential implications of interpreting the orbit of S2?

Interpreting the orbit of S2 has significant implications for our understanding of the universe. It provides evidence for the existence and properties of supermassive black holes, which play a crucial role in the formation and evolution of galaxies. It also supports the validity of general relativity and helps us to better understand the fundamental laws of physics.

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