# Why is pressure broadening greater in low mass stars than in high mass stars?

I have a question about broadened absorption lines. In "Discovering the Universe" by Comins and Kaufmann, it is mentioned that cool main sequence stars have broader absorption lines than hot main sequence stars. I believe that this effect is due to pressure broadening. However, I don't know why pressure broadening (i.e. surface gravity) would be higher for the less massive stars. I've been agonizing over this for a few weeks and any help would be appreciated.

The surface gravity is higher in low mass stars because of the way radius scales with star mass (recall that cool main sequence stars are lower in mass than hot main sequence stars). On the main sequence, the radius of a star scales with the mass; so, a star that is twice as massive as another will be twice the size of it. Now, surface gravity is essentially the force of gravity exerted on the outermost layers of a star by the mass interior to those layers. Recall that the force of gravity is proportional to the inverse square of the distance of the outer layers from the star's centre, and thus to the inverse square of the radius, and also to the star's mass. But, from above, the radius of the star is proportional to its mass. When you put it all together, you find that the surface gravity is proportional to the mass of the star divided by the mass squared: this means that the surface gravity is inversely proportional to the mass of the star. In other words, the force of gravity and the way mass along the main sequence scales with radius conspire to produce a surface gravity that decreases with increasing star mass. This means that pressure broadening is more important in low mass main sequence stars than in high mass ones.

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