Dark Matter — Question 3

Question 3

Subtracting the luminous contribution requires subtracting squared speeds

At a given radius, a spherical toy model predicts luminous-matter circular speed 120 km/s while the observed speed is 200 km/s. Independent gravitational potentials add, so $v_{\mathrm{total}}^2=v_b^2+v_h^2$.

1. Find the halo circular-speed contribution.

2. Find the luminous and halo fractions of enclosed dynamical mass under the spherical assumption.

Question 3 - Solution

Subtracting the luminous contribution requires subtracting squared speeds

(a) Method and calculation

$$v_h^2=200^2-120^2=25600\ ({\mathrm{km/s}})^2.$$$$\boxed{v_h=160\ {\mathrm{km/s}}}.$$

(b) Interpretation and further calculation

$$\boxed{f_b=M_b/M_{\mathrm{total}}=v_b^2/v_{\mathrm{total}}^2=.36}.$$$$\boxed{f_h=1-.36=.64}.$$

Physical check

Subtracting speeds directly would give 80 km/s and an incorrect mass decomposition. The result depends on the adopted luminous-mass model.