Question

Answer all of the following questions:

3. The galaxy cluster Abell 2218 has a mass of about 1.3x1015 Msun and a radius of about 1.5 Mpc. a. Estimate the free-fall time, t, for Abell 2218. b. The age of the universe is 13.8 Gyr. Is Abell 2218 gravitationally relaxed and why?

Answer 1

Given,

M = 1.3 x 10¹⁵ M_SUN = 1.3 x 10¹⁵ x (1.989 x 10³⁰ ) Kg = 2.59 x 10⁴⁵ Kg

R = 1.5 Mpc = 1.5 x (3.0857 x 10²²) m = 4.63 x 10²² m

G = 6.673×10^–11 kg^–1m³s^–2

(a) Free Fall Time, t_ff = (3 π /23G ρ ₀)^1/2

where, ρ ₀ = 3M / 4π R³ = 3 x (2.59 x 10⁴⁵ Kg) / 4 x 3.14 x (4.63 x 10²² m)³ = 1.357 x 10²²

Now,

Free Fall Time, t_ff = (3 π /23G ρ ₀)^1/2 = [3 x 3.14 / {23 x (6.673×10^–11 kg^–1m³s^–2) x (1.357 x 10²²)}]^1/2 = 6.72 x 10^-7 second

(b) Since the age of Universe is 13.8 Gyr = 13.8 x ( 3.15576×10⁺¹⁶s ) = 43.55 x 10⁺¹⁶ s