Question

A star near the visible edge of a galaxy travels in a uniform circular orbit. It is 41,200 ly (light-years) from the galactic center and has a speed of 275 km/s. Estimate the total mass of the galaxy based on the motion of the star.
Gravitational constant is 6.674×10−11 m3/(kg·s2) and mass of the Sun Ms=1.99 × 1030 kg.
*Answer in billion solar mass

Answer 1

The total mass of the galaxy is 443.4 Solar mass

Orbital velocity ($v$) = $\sqrt{\frac{MG}{R} }$

where M= weight of galaxy

G= gravitational constatnt = $6.674*10^-^1^1$ (given)

R = distance from centre = $41200$ Light years (given)= $4.12*9.5*10^1^6$  km (1 ly= $9.5*10^3$ billion km)

v= orbital velocity = $275$  $km/s$ (given)

∴ According to the formula

$(2.75*10^2)^2$ = $\frac{M*6.674*10^-^1^1}{4.12*9.5*10^1^6}$

⇒ $7.56*10^4*4.12*9.5*10^1^6=M*6.674*10^-^1^1$ (cross multiplying and expanding)

⇒ $29.59*10^2^1=M*6.674*10^-^1^1$

⇒ $\frac{29.59*10^2^1*10^1^1}{6.674}=M$

⇒ $4.434*10^3^2=M$

1 solar mass = $1.989*10^3^0 kg$

⇒ Mass in solar mass =443.4 Solar mass

⇒ M = 443.4 Solar mass

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