PHYSICS 428
INTRODUCTION TO ASTROPHYSICS
PROBLEM ASSIGNMENT #10
DUE: MONDAY, MARCH 31, 2014
The Distribution of Rotational Velocities, Pv(v) (10 points possible)
The distribution of vsin i , where v is equatorial speed of rotation and i is the angle between the axis of
rotation and line of sight, can, in principle, be determined observationally for any class of stars. We shall
represent that distribution by the function Pv sin i ( y) , where Pv sin i ( y)dy represents the probability that vsin i is
between y and y + dy. The more useful distribution function Pv (v) , the distribution function for the
equatorial speed of rotation is, unfortunately, not observable.
However, although Pv (v) cannot be directly observed, it is constrained by the observable
function Pv sin i ( y) . In class we derived the relationship between the two,
 y
Pv  dx
1
x
Pv sin i ( y )     ,
0
1 x2
which Pv (v) must satisfy. To check the validity of this expression, confirm that it does lead to the two
necessary results below:
(1) Show that this expression implies that Pv sin i ( y) is properly normalized, i.e., show that

0
Pv sin i ( y )dy  1
if and only if


0
Pv (v)dv  1 .
(2) Also show that this expression implies that the mean value of vsin i , v sin i , is related to the mean
value of v, v , by the relationship
v sin i 
as we proved in class by other means.

4
v ,