Using Newton's Version of Kepler's Third Law II The Sun orbits the center of the Milky Way Galaxy every 230 million years at a distance of 28,000 light-years. Use these facts to determine the mass of the galaxy. (As we'll discuss in Chapter Dark Matter, Dark Energy, and the Fate of the Universe, this calculation actually tells us only the mass of the galaxy within the Sun's orbit.) M = solar billion years

The mass of the galaxy is 2.096 * 10⁴¹ kg

Explanation:

Newton's Version of Kepler's Third law of motion II is:

p² = 4π² a³ / G (M + m) (1)

where

p is the orbital period a is the average distance between the sun and the galactic centre G is the universal gravitational constant M is the mass of the galaxy m is the mass of the sun

Step 1:

The orbital period of the sun around the galaxy is:

p = 230*10⁶ years * (3.15*10⁷ s / 1 year)

p = 7.25 * 10¹⁵ s

Step 2:

The average distance between the sun and the galactic centre is:

a = 28000 light-years * (9.46*10¹⁵ m / 1 light-year)

a = 2.65*10²⁰ m

Step 3:

Substitute the values of p and a into equation (1):

Rearranging equation (1) to make M the subject of the formula, we get:

M = (4π² a³ / G p²) - m

M = (4π² (2.65*10²⁰ m) ³ / (6.67*10⁻¹¹ m) (7.25 * 10¹⁵ s) ²) - 1.9891 * 10³⁰ kg

M = 2.096 * 10⁴¹ kg

Therefore, the mass of the galaxy is 2.096 * 10⁴¹ kg

mass of the galaxy = 1.05 * 10^11 solar masses

Explanation:

According to Kepler's third law, A^3 = P^2

Where A = Average distance of a planet from the sun, in AU

And P = The time taken by the planet to orbit the sun, in years.

Newton's modification to Kepler's third law applies to any two objects orbiting a common mass

According to Newton, M1 + M2 = (A^3) / (P^2)

Where M1 and M2 are the masses of the two objects in Solar mass

From the question,

Let M1 = the mass of the sun

and M2 = the mass of the milky way galaxy

Distance, A = 28,000 light years

1 light year = 63241.1 AU

A = 28000 * 63241.1

A = 1,770,750,800 AU

Time taken for the orbit, P = 230,000,000 years

M1 = 1 solar mass

M2 = ?

Using M1 + M2 = (A^3) / (P^2)

1 + M2 = (1770750800^3) / (230,000,000^2)

1 + M2 = 1.05 * 10^11

M2 = (1.05 * 10^11) - 1

M2 = 1.05 * 10^11 solar masses