Calculating the Mass of the Galactic Center with Kepler's Third Law

In summary, the Sun's speed in its orbit around the Galactic center is 220 km/sec and its distance from the center is 8500 pc. Using Kepler's Third Law, we can calculate the mass of the galaxy interior to the Sun's orbit. This is done by equating the force of the galaxy's mass on the Sun to the centrifugal force. One can then solve for the mass of the galaxy by substituting in the values for the distance and velocity of the Sun. However, there may be some confusion as to the relevance of Kepler's Third Law in this calculation, as it only describes the relationship between orbital periods and semimajor axes and does not take into account gravitational force or mass.
  • #1
Morpheus
I'm not good at this stuff, so I need some help from some smarter people...:smile:

The speed of the Sun in its orbit about the Galactic center is 220 km/sec. Its distance from the center is 8500 pc. Assuming that the Sun's orbit is circular, calculate the mass of the Galaxy (in solar masses) interior to the Sun's orbit from Kepler's Third Law. Recall that the circumference of a circle is 2 pi r, where r is the radius. Remember that 1 pc is 206265 AU and 1 AU is 1.5x10^8 km.

Any help would be greatly appreciated!
 
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  • #2
...

Assuming that the mass of the galaxy is concentrated at its center, the force of the galaxy of mass M on the sun of mass m separated by a distance r from the galaxy center is:

F = GMm/r^2. This force will be equal to the centrifugal force due the revolution of the sun around the galaxy.
Thus,

GMm/r^2 = mV^2/r, where, V is the velocity of sun around the galaxy.

From this u can calculate the value of M - the mass of the galaxy (Its, just mere substitution...)


Sridhar
 
  • #3
sridhar_n's response is exactly what you need but I am puzzled about the reference to "Kepler's third law".

Kepler's third law is: The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semimajor axes.

Of course one can extend this to stars orbiting the galaxy but this says nothing about mass. Kepler's laws were purely descriptive. Kepler knew nothing about gravitational force or how mass affected orbital period.
 
  • #4
Thanks for the help. All I know is my Astronomy professor asked us this question and told us to refer to Kepler's Third Law.
 

1. How does Kepler's Third Law help in calculating the mass of the Galactic Center?

Kepler's Third Law states that the square of a planet's orbital period is directly proportional to the cube of its semi-major axis. By observing the orbital period and distance of stars orbiting around the Galactic Center, we can use this law to calculate the mass of the center.

2. What data is needed to apply Kepler's Third Law to calculate the mass of the Galactic Center?

To apply Kepler's Third Law, we need to know the orbital period of stars around the Galactic Center, as well as their distance from the center. This data can be obtained through observations using telescopes and other astronomical instruments.

3. Can Kepler's Third Law be used to accurately calculate the mass of the Galactic Center?

Yes, Kepler's Third Law has been proven to be accurate in calculating the masses of celestial bodies, including the Galactic Center. However, other factors such as the presence of dark matter may affect the accuracy of the calculation.

4. Are there any limitations to using Kepler's Third Law for calculating the mass of the Galactic Center?

One limitation is that Kepler's Third Law assumes that the orbital motion is due to the gravitational pull of a single central body. In the case of the Galactic Center, there are multiple massive objects that may affect the orbital motion, making the calculation more complex.

5. How is the calculated mass of the Galactic Center used in scientific research?

The mass of the Galactic Center is an important factor in understanding the structure and evolution of our galaxy. It can also be used in studies of dark matter and the formation of supermassive black holes. Furthermore, it provides valuable information for future space missions and exploration of the Milky Way.

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