Question

Globular clusters are spherical clusters of stars that orbit a galaxy, some of the older ones in the Milky Way are 13 Gyr old. At what distance would these globular clusters cease to spiral into the galactic nucleus due to the effects of dynamical friction if it has a mass of 6×10⁶M⊙ and an orbital speed of 230 km/s?

Answer 1

given

t_max = 13 Gyr ( 10⁹ yr / Gyr )

= 13 x 10⁹ x 3.14 x 10⁷

= 40.8 x 10¹⁶ sec

M = 6 x 10⁶ M_⊙

= 6 x 10⁶ x 2 x 10³⁰

= 1.2 x 10³⁷ kg

v_m = 230 km/sec

= 230000 m/sec

C = 76

G = 6.67 x 10^-11 m³ / kg sec²

using equation

r_max = ( t_max C G M / 2 v_m )^1/2

= ( 40.8 x 10¹⁶ x 76 x 6.67 x 10^-11 x 1.2 x 10³⁷ / 2 x 3.14 x 230000 )^1/2

= 1.31 x 10²⁰ m

or

= 1.31 x 10²⁰ / 3.1 x 10¹⁹

= 4.225 kpc

At 1.31 x 10²⁰ m or 4.225 kpc distance would these globular clusters cease to spiral into

the galactic nucleus due to the effects of dynamical friction if it has a mass of 6 x 10⁶M_⊙

and an orbital speed of 230 km/s.