MadSci Network: Astronomy

Re: What are the equations that relate Mass and Temp. etc. in the HR Diagram

Date: Sat Apr 8 01:13:05 2000
Posted By: Jeff Brown, Faculty, Astronomy
Area of science: Astronomy
ID: 954858911.As

A few of the equations are simple, assuming you're going to try doing a "theoretical" Hertzsprung-Russell diagram, (another place to look is here) that is, effective temperature running backwards across the bottom (that is, high temperature towards the left) and luminosity (usually in solar units) up the side. These are always plotted with logarithmic scales.

On the other hand, I don't think that a complete H-R diagram is going to be that easy to plot. There are some decent empirical approximations for how some stars behave, but generally a star's luminosity and temperature are not anything like simple functions of initial mass, chemical composition, and time. For main sequence stars there is a handy relation between mass and luminosity; roughly luminosity goes as the 3.5 power of the mass of the star.

While this gets you a luminosity for a restricted class of stars (the admittedly very important and most-common main sequence stars), it doesn't get you the surface temperature, which is the other variable in the H-R diagram. I am afraid you are going to find almost nothing to let you relate mass to temperature, even for main sequence stars where there is at least some correlation between those. That's because while luminosity ends up being mostly dependent on stellar mass, there are a number of other things besides mass that go into determining a star's temperature.

If somehow you cobble up a temperature (and you might get one from a mass-luminosity relation and a mass-radius relation, and playing with those might get you a hokey mass-temperature relation) for stars, then you can use that for a "theoretical" diagram. If you want go from a temperature to a color or a spectral type (and these are the observational horizontal axis for an H-R diagram, and this is the form that came first), that's a much more difficult question; this is always done empirically and such improvements in such "color-temperature relations" are of ongoing interest to people like me. This is addressed very briefly here and here, though probably not to the level you'd like.

There's an instructional exercise where you can look for empirical patterns among stars here; this looks promising from my point of view in showing some of the difficulties involved but since I don't have Excel readily available to me that's a guess on my part.

It's worth pointing out that for post-main-sequence stars, giants, white dwarfs, and supergiants, the task of constructing H-R diagram position as function of mass is hopeless. There are more variables involved, the most important of which are time and chemical composition, which have to be specified before you can map a star to its position in the H-R diagram. This is brings in all the processes and problems of stellar evolution, and that's a much larger subject than you can get into in a simple fashion.

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