Consider a star with mass equal to our sun, 1M sun =3.3*10 5 M earth , orbiting
ID: 1487023 • Letter: C
Question
Consider a star with mass equal to our sun, 1Msun=3.3*105 Mearth, orbiting the center of the galaxy with mass Mgalaxy= 1017Mearth at a distance of r=3.1 *1010Rearth
3.1 What is the gravitational force between the star and galaxy?
3.2 Assume the orbit of the star about the galactic center is circular, Calculate the speed at which the star orbits the galaxy. Hint: the motion is circular.
3.3. If astronomers discover that the speed of the star is twice as large as they calculate, how massive must the galaxy be to account for the large speed? (hint: there is a quick way to solve his and a long way)
Explanation / Answer
We know mass of eath, Me = 5.98*10^24 kg
so,
mass of sun, Msun = 3.3*10^5*5.98*10^24
= 1.9734*10^30 kg
distance, r = 3.1*10^10*6.37*10^6 m
= 1.9747*10^17 m
mass of galaxy, Mgalaxy = 10^17*5.98*10^24
= 5.98*10^41 kg
3.1) the gravitational force between the star and galaxy, F = G*Mgalaxy*Msun/r^2
= 6.67*10^-11*5.98*10^41*1.9734*10^30/(1.9747*10^17)^2
= 2.02*10^27 kg <<<<<<<<-----------Answer
3.2) Orbital speed, vo =sqrt(G*Mgalaxy/r)
= sqrt(6.67*10^-11*5.98*10^41/(1.9747*10^17))
= 1.42*10^7 m/s <<<<<<<<-----------Answer
3.3) v2/v1 = sqrt(M2/M1)
(v2/v1) = M2/M1
M2 = M1*(v2/v1)^2
= M1*(2*v1/v1)^2
= 4*M1
M_actual = 4*M_calculated
Mgalaxy(actua) = 4*Mgalaxy(calculated)
= 4*5.98*10^41
= 2.392*10^42 <<<<<<<<-----------Answer