Respuesta :
Answer:
4.50 × 10² [tex]\textup{billion\ M}_s}[/tex]
Explanation:
Given:
Velocity of orbiting = 420 km/s = 4.2 × 10⁵ m/s
Distance = 11 kpc = ( 11 × 3.086 × 10¹⁹) m = 3.394 × 10²⁰ m
now,
the cluster is orbiting due to the balancing of the centripetal force and the force due to the gravitational pull by the galaxy
thus,
we have
force due to the centripetal force = [tex]F_c=\frac{mv^2}{r}[/tex] ............(1)
where,
m is the mass of the cluster
v is the velocity of the cluster
r is the distance between the galaxy and the cluster
also,
the force due to the gravitational pull = [tex]F_g=\frac{GMm}{r^2}[/tex] .......(2)
here,
G is the gravitational force constant = 6.67 × 10⁻¹¹ Nm⁻²kg⁻²
M is the mass of the galaxy
on equating the equation (1) and (2), we have
[tex]\frac{mv^2}{r}=\frac{GMm}{r^2}[/tex]
or
[tex]{v^2}=\frac{GM}{r}[/tex]
on rearranging and substituting the values, we get
M = [tex]\frac{3.39\times10^{20}\times(4.2\times10^5)^2}{6.67\times10^{-11}}[/tex]
or
M = 8.96 × 10⁴¹ kg
Now,
1 Solar mass [tex]\textup{M}_s}[/tex] = 1.99 × 10³⁰ kg
thus,
1 kg = [tex]\frac{1M_s}{1.99\times10^{30}}[/tex]
Hence,
8.96 × 10⁴¹ kg = 8.96 × 10⁴¹ kg × [tex]\frac{1M_s}{1.99\times10^{30}}[/tex]
or
= 4.50 × 10¹¹ [tex]\textup{M}_s}[/tex]
or
= ( 4.50 × 10² ) × 10⁹ [tex]\textup{M}_s}[/tex] = 4.50 × 10² [tex]\textup{billion\ M}_s}[/tex]
