Detailed Mass-Luminosity Relation for Main Sequence Stars

[Jenab]

Does anybody know of a decent curvefit for the exponent in the main sequence mass-luminosity relationship?

L/Lsun = (M/Msun)^a

One constant fits all (e.g., 3.5) doesn't seem to be good enough. I don't like piecewise discontinuous approximations because you'd get two different answers for the luminosity at the masses where one piece ends and the next one begins.

Somebody's probably worked out a curvefit for the parts of the main sequence for which the ML relation is known, but I can't seem to find it anywhere.

Anybody know?

Jerry Abbott

 

[Tao]

I would also like to know this. I have been searching the internet and have only found a few more detailed approximations. Even these few approximations do not agree.

An example of disagreements are the following:

one site said:

The net result is that the luminosity of a star grows disproportionately with increasing mass. The detailed dependence of luminosity depends on the mass range of the star.

For stellar masses up to about 1.5 solar masses. The luminosity grows proportionate to the fifth power of the stellar mass so that a 1 solar mass star has a luminosity 25 = 32 times the luminosity of a 0.5 solar mass star.

For medium mass stars between 1.5 and 3.0 solar masses, the luminosity grows proportionate to the third power of the stellar mass. So a 3 solar mass star has 23 = 8 times the luminosity of a 1.5 solar mass star.

For higher mass stars, the luminosity grows proportionate to the mass of the star. So a 10 solar mass star has twice the luminosity of a 5 solar mass star.

While anothe rsite told me something like masses close to one sm (I can't remember the range(something like .5 to 5 sm)) are repersented by L = M^4, while mass above and below the range are repersented by L = M^3.3. The two above examples make me wonder which is closer.

I believe they are only accurate for masses closer to the solar mass (if you consider around .5 to 10 sm close). This is annoying for me since I am trying to construct a pseudoscientific model of a galaxy for a game I am trying to create. Also I need to know a function that relates mass to the probability of the star occurrence. Finally, although I probably need to search on my own more, I need to know the average distance between stars (6 to 7 ly) including the standard deviation.

 

[Tao]

This link may help

http://aa.springer.de/papers/9341001/2300121/sc6.htm

Although I did not take a close look, I believe it suggests that, at least so far, there is no "accurate" mass-luminosity relation. The mass-luminosity varies greatly to some extent and is, therefore, only an approximation. Even this page (the link I provide) is, I believe, only looking at masses ranging form .5 to 2 solar masses.

So the jumps in "piecewise discontinuous approximations" may not be as bad as you thought, or I. Still I would like a better approximation then L = M^n
n being between 3 and 4. Importantly for me, I would like the relation to be accurate for mass from .008 to 100 solar masses (although I wonder whether the accuracy drops with the increase of mass).

 

[Jenab]

I found something at
http://www.phys.unm.edu/~duric/phy536/3/node2.html

For M/Msun < 0.43, b = 0.23 and a = 2.30

For M/Msun > 0.43, b = 1.00 and a = 4.00

where L/Lsun = b (M/Msun)^a

That sounds plausible for M/Msun below about 3.0. Above there, the exponent (a) seems to trail off slowly to about 3.4.

There's a graph of data points at
http://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Unit2/mlrel.gif
(Bad color scheme though; the green used for the points is too light.)

You could fit your own log-log lines by measuring with a rular, then convert to the exponent form.

There might be some difficulty getting the ML relation for very low mass main sequence stars because their dimness makes it hard to observe many eclipsing binaries and also because the flares of some of them have a higher brightness in proportion to the usual luminosity of the star itself.

Jerry Abbott

 

[Jenab]

This link may help

http://aa.springer.de/papers/9341001/2300121/sc6.htm

Although I did not take a close look, I believe it suggests that, at least so far, there is no "accurate" mass-luminosity relation. The mass-luminosity varies greatly to some extent and is, therefore, only an approximation. Even this page (the link I provide) is, I believe, only looking at masses ranging form .5 to 2 solar masses.

So the jumps in "piecewise discontinuous approximations" may not be as bad as you thought, or I. Still I would like a better approximation then L = M^n
n being between 3 and 4. Importantly for me, I would like the relation to be accurate for mass from .008 to 100 solar masses (although I wonder whether the accuracy drops with the increase of mass).

Yes, the composition is different. Some stars have more metals than others do, and that might make the main sequence ML diagram fuzzy. But it's worth going after the detailed ML relation for the Population I star of average composition, like the sun is maybe.

The star mass low bound is 0.08, I think. I forgot what it's called. The Chandrasakar mass maybe?

Jerry Abbott

 

[Tao]

Yeah, .08 not .008. My mistake.

On further a look I found that m^4 is bad for high mass stars. According to my information stars with solar masses around 6 sm correspond to close to 300 L. m^4 gives a luminosity of nearly 1300. This is a little over 4 times the luminosity that my sources give me. I will try approximating like you suggested and, perhaps, check my sources.

 

[Jenab]

Mass Luminosity Relation (from data on graph)

I eyeballed these points for drawing lines between.

point 1: M=0.1, L=0.0012
point 2: M=0.2, L=0.0052
point 3: M=0.4, L=0.025
point 4: M=1.0, L=1.0
point 5: M=1.42, L=4.0
point 6: M=2.35, L=49
point 7: M=5.9, L=1200
point 8: M=42, L=41000

Taking logarithms:

point 1: -1.000, -2.921
point 2: -0.699, -2.284
point 3: -0.398, -1.602
point 4: 0, 0
point 5: +0.152, +0.602
point 6: +0.371, +1.690
point 7: +0.771, +3.079
point 8: +1.623, +4.613

P = log M
Q = log L

point 1 to point 2: Q = 2.116 P - 0.805
point 2 to point 3: Q = 2.266 P - 0.700
point 3 to point 4: Q = 4.025 P + 0.000
point 4 to point 5: Q = 3.960 P + 0.000
point 5 to point 6: Q = 4.968 P - 0.153
point 6 to point 7: Q = 3.472 P + 0.402
point 7 to point 8: Q = 1.800 P + 1.691

Linear fit mass-luminosity relations:

[M < 0.2], b=0.157, a=2.116
[0.2 < M < 0.4], b=, a=2.266
[0.4 < M < 1.0], b=1, a=4.025
[1.0 < M < 1.42], b=1, a=3.960
[1.42 < M < 2.35], b=0.703, a=4.968
[2.35 < M < 5.9], b=2.523, a=3.472
[5.9 < M < 42], b=49.09, a=1.800

L = b M^a

The high coefficient (49.09) tells me that I probably aimed too low on the bright end of the brightest segment of the main sequence. I wonder whether the jump in the exponent (a) from M=1.42 to M=2.35 might have something to do with the C-N-O fusion cycle starting to dominate proton-proton.

A quadratic log curvefit might be better. Smoother, anyway. More likely to please the critical people who go number-crunching after science-fiction authors.

P = log M

point 1 to point 3 (mass under 0.4 solar masses):
Q = +0.24853 P^2 + 2.5385 P - 0.6310

point 3 to point 5 (mass from 0.4 to 1.42 solar masses):
Q = -0.11746 P^2 + 3.9784 P + 0.0000

point 5 to point 7 (mass from 1.42 to 5.9 solar masses):
Q = -2.4160 P^2 + 6.2316 P - 0.2894

L = 10^Q

I don't have any good guesses for the luminosity above ~6 solar masses yet.

Here are some points from the quadratic log fit.

M, L

0.1, 0.0012
0.2, 0.0052
0.3, 0.0129
0.4, 0.0250
0.5, 0.0619
0.6, 0.1293
0.7, 0.2404
0.8, 0.4105
0.9, 0.6572
1.0, 1.0000
1.1, 1.4604
1.2, 2.0620
1.3, 2.8300
1.4, 3.7918
1.5, 5.4077
1.6, 7.6196
1.7, 10.432
1.8, 13.929
1.9, 18.195
2.0, 23.311
2.5, 64.241
3.0, 136.09
3.5, 243.14
4.0, 386.06
4.5, 562.71
5.0, 769.02
5.5, 999.87
6.0, 1249.7

Jerry Abbott

 

[Tao]

I just thought of something. One could try to find an best fit equation or equations that relates mass to radius. I believe luminosity is both strongly related to mass and radius. Using this information the complexity of the equation would increase and, perhaps be more accurate.

Also, if you are interested *shrug* data points for mass, luminosity, and radius can be found at:

http://www.phy.umist.ac.uk/Teaching/PastExams/MoreExams2/00-2/3/P611_99-00/node2.html

and

http://curriculum.calstatela.edu/courses/builders/lessons/less/les1/StarTables_B.html

 

[Jenab]

There might not be enough data for a bivariate model L(M,R). And R seems to be a fairly tight function of M, too. For most of the main sequence,

R = M^0.72

Or something like that.

If...

L = M^a

and

R = M^b

then the effective temperature is

T = 5770 Kelvin M^(a/4 - b/2)

You can develop equations for the star's average density and time-on-main-sequence, too.

Jerry Abbott

 

[Jenab]

http://www.phy.umist.ac.uk/Teaching/PastExams/MoreExams2/00-2/3/P611_99-00/node2.html

and

http://curriculum.calstatela.edu/courses/builders/lessons/less/les1/StarTables_B.html

Whoever made those websites used different data than I used. If I had an observatory, I'd probably go after definitive answers. There's too wide a variation in the M-L relation for it to be a matter of composition differences only.

Jerry Abbott

 

[Tao]

I am not sure if this is helpful but I came across the following from a Russian astronomy journal:
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=ATROES000042000006000793000001&idtype=cvips&gifs=yes&jsessionid=2345211086884141292

I believe that in the above article Msub(bol) mean the bolometric magnitude.

 

[Tao]

Perhaps there is not enough data like you said or it would not be any more accurate but doesn't luminosity equal 4piR^2T^4 ? Where pi is 3.14..., R is the radius, and T is temperature. If there was enough data one could relate mass and radius, and also relate mass and temperature using an approximation method. In relating them you would put them in the following forms: R(M) and T(M). Substituting R(M) for radius and T(M) for temperature in the above luminosity relation would give you L(M). Perhaps this new approximation would be more accurate.

 

[Jenab]

Perhaps there is not enough data like you said or it would not be any more accurate but doesn't luminosity equal 4piR^2T^4 ? Where pi is 3.14..., R is the radius, and T is temperature. If there was enough data one could relate mass and radius, and also relate mass and temperature using an approximation method. In relating them you would put them in the following forms: R(M) and T(M). Substituting R(M) for radius and T(M) for temperature in the above luminosity relation would give you L(M). Perhaps this new approximation would be more accurate.

The relations L(M) and R(M) are determined empirically from observations of eclipsing binary stars. Once they are known, you can use the Stephan-Boltzmann relation to get T(M) because it defines a star's effective temperature. This is why the effective temperature is distinct from color temperature, which is based on U-B and B-V color excess, although the two are usually fairly close.

Jerry Abbott