Question

A binary star system consists of two equal mass stars that revolve in circular orbits about their center of mass. The period of the motion, T=25.5 days and the orbital speed ????=220 km/s of the stars can be measured from telescopic observations. What is the mass of each star?

Answer 1

Answer:

$m = 2.23 \times 10^{-32} kg$

Explanation:

Given data:

PERIOD OF MOTION IS T = 25.5 days

orbital speeds = 220 km/s

we know that

acceleration due to centripetal force is$a =   \frac{F}{m} = \frac{V^2}{r}$

Gravitational force$F= \frac{Gm m}{d^2}$

we know that

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

solving for

$R = \frac{vT}{2\pi}$

$F = \frac{Gm^2}{4(\frac{vT}{2\pi})^2}$

$F = G\times \frac{\pi m}{(vT)^2}$

$a = \frac{v^2}{\frac{vT}{2\pi}}$

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

we know that

f =ma

$G\times \frac{\pi m}{(vT)^2} = a = \frac{2\pi m v}{T}$

solving for m

$m = \frac{2Tv^3}{\pi G}$

$m = \frac{2\times 25.5 \times 86400 \times 220000^3\ m/s}{\pi \times 6.67\times 10^{-11}}$

$m = 2.23 \times 10^{-32} kg$

Answer 2

Based on the calculations, the mass of each star is equal to $2.24 \times 10^{26}\;kg$

Given the following data:

Period = 25.5 days.

Orbital speed = 220 km/s.

Scientific data:

Gravitational constant = $6.67 \times 10^{-11}$

Conversion:

Period = 25.5 days to seconds = $25.5 \times 24 \times 3600$ = 2203200 seconds.

Orbital speed = 220 km/s to m/s = $220\times 10^3$ m/s.

How to calculate the mass of each star.

From Newton's law of motion and the law of universal gravitation, the mass of a planetary object is given by this formula:

$m=\frac{2TV^3}{\pi G}$

Where:

• T is the period.

• V is the orbital speed.

• G is the gravitational constant.

Substituting the given parameters into the formula, we have;

$m=\frac{2 \times 2203200 \times (220\times 10^3)^3}{3.142 \times 6.67 \times 10^{-11}}\\\\m=\frac{469193472 \times 10^{14}}{2095714 \times 10^{-10}} \\\\m=2.24 \times 10^{26}\;kg$

Read more on orbital speed here: https://brainly.com/question/4854338