The solar system is 25,000 light years from the center of our Milky Way galaxy. One light year is the distance light travels in one year at a speed of 3.0 x 10⁸ m/s. Astronomers have determined
that the solar system is orbiting the center of the galaxy at a speed of 230 km/s.
a. Assuming the orbit is circular, what is the period of the solar
system’s orbit? Give your answer in years.
b. Our solar system was formed roughly 5 billion years ago. How many orbits has it completed?
c. The gravitational force on the solar system is the net force due to all the matter inside our orbit. Most of that matter is concentrated near the center of the galaxy. Assume that the matter has a spherical distribution, like a giant star. What is the approximate mass of the galactic center?
d. Assume that the sun is a typical star with a typical mass. If galactic matter is made up of stars, approximately how many stars are in the center of the galaxy?

Astronomers have spent many years trying to determine how many stars there are in the Milky Way. The number of stars seems to be only about 10% of what you found in part d. In other words,
about 90% of the mass of the galaxy appears to be in some form other than stars. This is called the dark matter of the universe. No one knows what the dark matter is. This is one of the outstanding scientific questions of our day.

Step by step solution

01

Part(a) Step1: Given information

Radius of the solar system, r=25000 light year
Speed v=230 km/s

02

Part(a) Step2: Explanation

Calculate time period by

$T=\frac{2\pi r}{v}$

where v =velocity and r = radius

Substitute the value and calculate

$$\begin{gathered} T=\frac{2\pi \times (25000\times 3\times {10}^8\times 365\times 24\times 60\times 60m)}{(230\times {10}^{3}m/s)} \\ =6.46\times {10}^{15}\sec \times \left(\frac{1\text{year}}{365\times 24\times 60\times 60\sec }\right) \\ =2.05\times {10}^8\text{years} \end{gathered}$$

Period is 2.05 x 10⁸ years

03

Part(b) Step1: Given information

Average life of the solar system, Tⁿ=5 billion years =5 x 10⁹ years

04

Part(b) Step2 : Explanation

Number revolutions completed can be calculated as

$$\begin{gathered} N=\frac{T^n}{T} \\ =\frac{5\times {10}^{9}years}{2.05\times {10}^{8}years} \\ =24.4 \end{gathered}$$

Number of revolution completed is 24.4.

05

Part(c)Step1: Given information

Radius of the solar system, r=25000 light year
Speed v=230 km/s

06

Part(c) Step2 Explanation

We can calculate mass of the giant star as

$M=\frac{v^{2}r}{G}$

Substitute values, we get

$$\begin{gathered} M=\frac{{\left(230\times {10}^{3}m/s\right)}^2\times (25000\times 3\times {10}^8\times 365\times 24\times 60\times 60m)}{(6.67\times {10}^{-11}N/k{g}^2.m^2)} \\ =1.87\times {10}^{41}\mathrm{kg} \end{gathered}$$

Mass of the star is 1.87 x 10⁴¹ kg.

07

Part(d)Step1: Given information

Mass of sun, M_s=1.99 x 10³⁰ kg

08

Part(d)Step2:Explanation

Number of the star can be calculated as

$$\begin{gathered} N=\frac{\text{Mass of glactic center}}{\text{mass of sun}} \\ N=\frac{1.88\times {10}^{41}}{1.99\times {10}^{30}} \\ N=9.45\times {10}^{10} \end{gathered}$$

Number of starts is 9.45 x 10¹⁰ which is around 94.5 billion

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