Relationship betwen stellar mass loss and mass of stars

Question

I am following the semi-empirical relationship of mass loss rate of a star with mass M, radius R and luminosity L.

$log(\dot{M}v_{inf}R^{1/2})=−1.37+2.07log(L/10^{6})$

where $v_{inf}=\frac{\sqrt{(2GM/R)}}{2.6}$

M, R and L are in solar unit.

I simplified the relationship for plotting purpose:

$log(\dot{M})=−1.37+2.07log(L)−2.07log(10^{6})−\frac{1}{2}log(2*G*M)+log(2.6)$

I hope my simplification is correct. If not, please direct me to the right path.

I need to plot a $log(\dot{M})−M$ relationship for mass range $20−100M_{\odot}$ for few different luminosities and compare it with the main sequence $log(\dot{M})−M$ relationship.

My first confusion came with the value of G. Since everything is in solar unit, I thought I need to set everything with respect to solar unit.

So I took $G=4 \pi^{2} M_{\odot}^{-1} Au^{3} yr^{-2}$

.

I am not quite sure if this is the correct way to set G and L, so I plotted the data, and it looks like following:

But I was expecting higher mass loss for more massive stars. Please help me understand what am I doing wrong here.

Answer 1

There's nothing wrong with your plot. What's incorrect is your expectation that as you hold the luminosity constant, higher-mass stars should have higher mass loss.

Holding luminosity constant, higher-mass stars have higher escape velocities, which means it's more difficult for gas to escape their gravity well. As such, the expectation is that less mass loss should occur, which is exactly what your plot shows.

In reality, more massive stars typically have higher mass loss because more massive stars are usually much more luminous than their less massive brethren. This increase in luminosity offsets the increased depth of their gravity well, and leads to a net increase in mass loss. For example, as you go up the main sequence in mass, luminosity also increases, at a rate faster than the mass.