# Mass-Luminosity relationship

jdlawlis
In one of my textbooks, the authors claim that luminosity depends on the cube of the mass, yet several online resources say that luminosity varies as mass to the fourth power. Which one is correct?

Helios
I doubt there's any such law, especially mass to any power. Jupiter has mass, so where is the luminosity?

matt.o
I doubt there's any such law, especially mass to any power. Jupiter has mass, so where is the luminosity?

There is for main sequence stars, though there are caveats. For example, the exponent is not constant across the entire mass range. https://www.physicsforums.com/showthread.php?t=29701" thread may help.

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jdlawlis
Thanks for your help. I should have clarified at the beginning that I was talking about main sequence stars.

Jenab6
Here's a mass luminosity relationship that fits the empirical data closely.

Lower main sequence (M < 0.6224)
log L = 2.5186 log M - 0.4814

Middle main sequence (0.6224 < M < 1.6959)
log L = -1.142866 (log M)^2 + 4.621390 log M

Upper main sequence (M > 1.6959)
log L = 3.8931 log M + 0.1069

http://zebu.uoregon.edu/~imamura/208/feb6/ml.gif [Broken]
(Fit to data is mine.)

The main sequence lifetime is found as

T = (10 billion years) M / L

where L = 10^(log L), and where for (log L) you substitute the function of log M from above. M is in units of the sun's mass. The sun's mass is 1.99E+30 kilograms. L is in units of the sun's luminosity. The sun's luminosity is 3.826E+26 watts.

Mass, Luminosity, Time on Main Sequence (10^9 years)
0.1, 0.00100, 1000.000
0.2, 0.00573, 349.024
0.3, 0.01591, 188.557
0.4, 0.03284, 121.818
0.5, 0.05760, 86.8049
0.6, 0.09117, 65.8112
0.7, 0.18060, 38.7600
0.8, 0.34786, 22.9975
0.9, 0.61114, 14.7265
1.0, 1.00000, 10.0000
1.1, 1.54644, 7.11310
1.2, 2.28435, 5.25314
1.3, 3.24892, 4.00133
1.4, 4.47621, 3.12765
1.5, 6.00272, 2.49887
1.6, 7.86503, 2.03432
1.7, 10.0939, 1.68418
1.8, 12.6096, 1.42748
1.9, 15.5638, 1.22078
2.0, 19.0038, 1.05242

Mass, Luminosity, Time on Main Sequence (10^6 years)
3.0, 92.1257, 325.64
4.0, 282.345, 141.67
5.0, 673.070, 74.286
6.0, 1368.74, 43.836
7.0, 2494.31, 28.064
8.0, 4194.88, 19.071
9.0, 6635.31, 13.564
10.0, 10000.0, 10.000
15.0, 48477.6, 3.0942
20.0, 148573, 1.3461

Mass, Luminosity, Time on Main Sequence (10^3 years)
25.0, 354177, 705.862
30.0, 720246, 416.524
35.0, 1.31e6,.266.659
40.0, 2.21e6, 181.209
45.0, 3.49e6, 128.881
50.0, 5.26e6, 95.0189

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turboguppy
I doubt there's any such law, especially mass to any power. Jupiter has mass, so where is the luminosity?

Jupiter gives off more energy than it receives from the sun: Luminosity.

Also, we're not talking about a LAW law, we're talking about a power law--since there's a strong correlation between the Log(Mass) and Log(Luminosity) of main sequence stars, we can apply an empirical power law to show this relationship. Of course this is not a full model, but when done correctly can actually mimic models quite well.

If you are still looking for a good curve fit, I just did one myself with the data found at:http://isthe.com/chongo/tech/astro/HR-temp-mass-table-bymass.html" [Broken]

It goes:
IF:--------------> Then:
M <= .43--------> L = 1.2*M^4 + .00184*M^-.2
.43 < M <= 2----> L = M^4
2 < M <= 20----> L = 1.5*M^3.53 -0.0252982*M^4.3
M > 20---------> L = -.128*M^3.5 + 63*M^2.3

Been working on this for a long time so I hope it helps, or if not, then I hope you can point me toward better data to match!

~Mike

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Staff Emeritus
Thanks for that list Janab6! Very interesting to see! I had no idea of how big a difference another 10% solar masses causes.

Gold Member
There's even a simple way to understand why L ~ M^3 for high-mass stars, if you will take a short breath first. A star is basically a leaky bucket of light, so to know its luminosity, all you need to know is its internal temperature and volume (that gives you the internal radiant energy), and how long it takes that light to leak out (that's what you need the opacity for). The time scales like the radius over the average light diffusion speed, and the latter turns out to be inversely proportional to the optical depth. High-mass stars have mostly free-electron opacity, which is constant per gram, so for constant opacity, the optical depth is proportional to the radius times density, and since density scales like mass over volume, the optical depth scales like M/R^2, so the diffusion speed scales like R^2/M, and the escape rate scales like speed over distance, so that's R/M. Now, it turns out that from the virial theorem, the R of a main sequence star is fairly proportional to the M, so the escape rate (and escape time) for the photons inside a main sequence star is pretty much constant.

Whew. So we have that photons always escape in more or less the same amount of time across the high-mass end of the main sequence. This is very key, because now we just need to know how much radiant energy is in a star. Massive main sequence stars are all at basically 20 million K core T, and the energy density in radiation depends only on T, so all high-M MS stars have about the same radiant energy density. Thus, their radiant energy content scales roughly with the volume, or R^3. So look what we have-- MS star energy content scales like the volume of the big leaky bucket of light, the light escape time is all about the same, so the luminosity scales like the volume. Because of the virial theorem, the R is proportional to M. Voila, luminosity scales like M cubed.

Incidentally, an enormous number of seemingly authoritative web sites will tell you that the luminosity of high-mass stars is set by details of the fusion process, which is complete baloney. The analysis I just gave never needed to mention fusion at all.

qraal
Here's a mass luminosity relationship that fits the empirical data closely.

Lower main sequence (M < 0.6224)
log L = 2.5186 log M - 0.4814

Middle main sequence (0.6224 < M < 1.6959)
log L = -1.142866 (log M)^2 + 4.621390 log M

Upper main sequence (M > 1.6959)
log L = 3.8931 log M + 0.1069

http://zebu.uoregon.edu/~imamura/208/feb6/ml.gif [Broken]
(Fit to data is mine.)

The main sequence lifetime is found as

T = (10 billion years) M / L

where L = 10^(log L), and where for (log L) you substitute the function of log M from above. M is in units of the sun's mass. The sun's mass is 1.99E+30 kilograms. L is in units of the sun's luminosity. The sun's luminosity is 3.826E+26 watts.

Mass, Luminosity, Time on Main Sequence (10^9 years)
0.1, 0.00100, 1000.000
0.2, 0.00573, 349.024
0.3, 0.01591, 188.557

A modifier on very low mass stars and their lifetimes is that they very efficiently convect and so all their hydrogen is available on the Main Sequence, unlike Sun-like stars which can't get at their non-Core hydrogen until after the Main Sequence. As a result M-dwarfs last several times longer than what you've computed. The details can be found in this paper...

http://iopscience.iop.org/0004-637X/482/1/420/35131.text.html"

...with a more recent discussion of the Galaxy's changing luminosity over distant cosmic time here...

...thus the Galaxy will shine with about the same brightness for the next 800 billion years thanks to the steady brightening of aging Red dwarfs.

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Gold Member
Interesting point, that makes sense.

twofish-quant
There's even a simple way to understand why L ~ M^3 for high-mass stars, if you will take a short breath first.

Question: Is there an order of magnitude estimate for low mass stars? If you assume that the core is mostly convective, can you get numbers out? Also can you match differences in the power law with different physics?

Massive main sequence stars are all at basically 20 million K core T, and the energy density in radiation depends only on T, so all high-M MS stars have about the same radiant energy density.

Interesting. Why is that? I'm guess that there is some feedback mechanism.

Incidentally, an enormous number of seemingly authoritative web sites will tell you that the luminosity of high-mass stars is set by details of the fusion process, which is complete baloney. The analysis I just gave never needed to mention fusion at all.

Interesting. That means that the physics of the main sequence is an opacity thing, that is independent of the fusion reactions.

Gold Member
Question: Is there an order of magnitude estimate for low mass stars? If you assume that the core is mostly convective, can you get numbers out? Also can you match differences in the power law with different physics?
That's a good question. In the vicinity of the Sun, you have a steeper dependence of L on M, yet convection is not yet dominant, so it's probably more about the complicated opacity in a cooler star. If you go to the lower masses, say <0.5 solar, you're right that convection does become dominant, and there is a general theory about how such stars behave, called the "Hayashi track." Adding the condition that nuclear fusion, rather than gravitational contraction as in protostars, is providing the luminosity, constrains the behavior on the Hayashi track. There might be a simple way to get luminosity under those conditions, I'd have to bone up on that. The problem is that convection is treated with "mixing length theory", which is an approximate approach that is pretty unwieldy and is used more for lack of an alternative than because of its dependability. So the short answer is, "yes you can estimate the luminosity of a convective main-sequence star of given M, but it's not easy, not reliable, and I don't recall how to do it."
Interesting. Why is that? I'm guess that there is some feedback mechanism.
Yes, it is due to the extreme T sensitivity of the fusion process, especially the CNO cycle in massive stars. Tiny increases in T produce huge increases in fusion rate, so this acts like a thermostat on T because it is stabilized by gas pressure (T rises cause adiabatic expansion which drops T back down).

Ironically, you very often see this extreme T sensitivity as evidence that the fusion rate must control the luminosity, but that is exactly backward. The high T sensitivity is the reason that the luminosity controls the fusion rate. Its T sensitivity makes the fusion rate easily manipulated by the factors in the star that actually do control its luminosity (its thermodynamic structure). Hence you never need to know anything about fusion other than that it acts like a 20 million K thermostat. It is certainly not true that the reason higher mass stars are more luminous is that their cores are hotter-- that is the backwards reasoning I'm talking about. (By the way, incredibly, you will also hear it stated that higher mass stars are more luminous because their higher pressure cores produce higher fusion rates. That's worse than backward, it is downright false-- higher mass stars have lower pressure cores, which is why they supernova.)
Interesting. That means that the physics of the main sequence is an opacity thing, that is independent of the fusion reactions.
Exactly. All you need fusion for is to be a thermostat, when coupled with the stabilizing effects of gas pressure. If you have an observational means of inferring the core T, you don't need fusion at all, which is how Eddington was able to model stars without knowing anything about fusion, he just needed on ad hoc element-- a characteristic value for the core T that had to be observed not explained.

turboguppy
Wow, I didn't expect the topic would revive so dramatically! Thanks for all the great info Ken G. I have been doing a lot of reading lately about how the internal structures of the stars (radiative zones vs convective) and opacity are what influence the Mass-Luminosity relationship at different mass values along the main sequence, but the only information I could find on higher mass stars was L $\propto$ M.

Since I'm only an amateur (VERY amateur) mathematician, I was getting over what that meant. I have encountered L $\approx$ M3 for high mass stars before, but seeing your explanation along side it helped it gel in my head.

I've been working for some time on a computer program that will generate/manage star systems for SF writers. I love learning about the mathematical relationships, but I really stink at math (unfortunate, I know), so I try to keep everything as simple as possible. Using equations that cancel out most of the constants is my favorite (sun = 1 is much more fun!) :rofl:

Finding a curve-fit piece-wise solution to the Mass - Luminosity relationship really made my day. Problem is, I can't seem to find information on masses and luminosities across the entire main sequence to compare. I am guessing this is because there are still lots of holes in the binaries data available for both high and low mass stars. Unfortunately, this general lack leads me to wonder how accurate the fits are. :yuck:

I have read there is a strong correlation between mass and radius as well, and if I can come up with a mass, radius and luminosity, I can then get a surface temperature (L ≈ R2*T4). Then I can assign spectral type, absorption lines, variability, metallicity, and age. Season with some spectral oddities here and there, and I should get a pretty nice approximation of what would pop out of a supercomputer left to crunch bigger equations for about a year.

So I have a few questions about stars for anybody who sounds like they know what they're talking about:

1. How close on the Mass - Luminosity and Mass - Radius relationships is 'Close Enough for Government Work?' Does anyone have any better suggestions across the range of masses?

2. How does a person like myself, trying to keep the math simple, account for age and metallicity effects on luminosity and radius? Age I have found something like a 7% increase per 100 million years, but for metallicity I haven't really found anything...

3. Is there an *easy* way to estimate mass loss over time for higher mass stars? An approximation, perhaps?

Many thanks to anyone who has answers. More thanks to anyone who can direct me to a good source for mass, radius, and luminosity values to play with.

twofish-quant
Season with some spectral oddities here and there, and I should get a pretty nice approximation of what would pop out of a supercomputer left to crunch bigger equations for about a year.

In fact stellar evolution codes generally take a few hours to run on a plain old workstation, and if you are handy with computers you can download this one...

http://mesa.sourceforge.net/

The reason that stellar evolution codes run quickly is that you can "average" the complicated physics over a large time step. So for example, you have complex turbulence and pressure effects, so you assume that those "average" out to a stable value over tens of thousands of years.

1. How close on the Mass - Luminosity and Mass - Radius relationships is 'Close Enough for Government Work?' Does anyone have any better suggestions across the range of masses?

Depends on how close you want things.

2. How does a person like myself, trying to keep the math simple, account for age and metallicity effects on luminosity and radius? Age I have found something like a 7% increase per 100 million years, but for metallicity I haven't really found anything...

In solar stars, metalicity doesn't change. All of the fusion is in the core, which means that the composition of the outer layers doesn't change much. What happens is that the core turns into helium and the helium will eventually turn into carbon.

Metalicity has a big impact between generations of stars, but most all of the metals are formed in supernova.

3. Is there an *easy* way to estimate mass loss over time for higher mass stars? An approximation, perhaps?

Topic of current research. For massive stars mass loss works as something of a fudge factor that you put in a model so that the models match observations. The other thing that makes things messy is if you have binary and multiple star systems, mass loss gets very complex.

turboguppy
In fact stellar evolution codes generally take a few hours to run on a plain old workstation, and if you are handy with computers you can download this one...

http://mesa.sourceforge.net/

The reason that stellar evolution codes run quickly is that you can "average" the complicated physics over a large time step. So for example, you have complex turbulence and pressure effects, so you assume that those "average" out to a stable value over tens of thousands of years.

I was of course referring to those simulations that are 3D and work through the creation of many, many stars starting with a massive cloud of gas. I don't remember how long the program I watched took to run, surely not a whole year, but I have been known to exaggerate a little to keep things light

It looks like MESA may be just a bit more complex than what I'm going for in my own program. I'm probably going to be generating entire 'sectors' worth (hundreds at least) of stars at once, so for my purposes empirical approximations are fine (and a lot faster). I'll also be making approximate models for proplyds, accretion, bombardments, migrations, and many other factors in the generation and placement of planets. This is a lot of calculating and iterating--I'd like to keep my calculations as simple as possible.

However, that said, MESA may be able to fill in some gaps in the data for me. Thanks for pointing me toward it.

Depends on how close you want things.

Good point. I am going for close enough to generally match observations. Working out a curve-fitting piecewise continuous power law calculation won't precisely model internal stellar physics, but it should be close enough to reflect what's going on.

I have several different examples of main sequence star properties (in the form of the little 'Spectral Type' charts that come on various astronomy websites), but few seem to agree on luminosity and radius values. From the best source of data I could find, I plotted M/L and M/R. From these plots I could see a single power law doesn't cover the full range of masses. After some reading I determined this was because of differences in structure, convection efficiency, opacity, etc.

The piecewise calculations I've found and/or modified fit the stellar data I found fairly well. Do these fits for the M/L and M/R relationships approximate what we observe closely enough that those who do stellar physics would feel comfortable using them to explain the relationships of stellar properties to the layperson? (that was a mouthful) If not, is there something better?

I also can't find a chart or explanation of how EXACTLY spectral types are assigned. I know it has to do with absorption lines in the spectra, and that surface temp correlates--but how does one wind up with a F2.5V star? Is it spectra to get the letter and then effective temp to get the number(and decimal place)?

In solar stars, metalicity doesn't change. All of the fusion is in the core, which means that the composition of the outer layers doesn't change much. What happens is that the core turns into helium and the helium will eventually turn into carbon.

Metalicity has a big impact between generations of stars, but most all of the metals are formed in supernova.

Well of course! I'm after the types of differences you would see in stars of differing starting metallicity. I keep seeing mention of ZAMS stars having such and such a mass-luminosity relationship, excluding metallicity effects. So the amounts of Fe/H in a star when it first contracts seem to have some effect on luminosity, and I'm guessing on radius as well. Determining what generation a star is, how much metallicity it has, also has implications for the planets in orbit around that star--such as how much raw materials are available to make rocky planets.

To clarify the question: For stars of different starting metallicity, how do I figure out the impact of + or - metallicity on ZAMS properties?

Topic of current research. For massive stars mass loss works as something of a fudge factor that you put in a model so that the models match observations. The other thing that makes things messy is if you have binary and multiple star systems, mass loss gets very complex.

Ha! Pretty familiar with the 'fudge factor' as I've also got to do atmospheric stuff like greenhouse effect and such. I'll keep digging on this, but it seemed to me that once you knew the mass and the expected luminosity for that mass, you should have a good idea if the star has to shed mass because of too much radiative pressure. Without modeling the star completely it seems I should be able to approximate or at least fudge a mass-loss rate based on the expected luminosity. Does that even make sense?

I'm planning to have some binaries in my results--the messy kind. I don't just want star systems that are perfectly habitable and completely stable. I want to generate some places SF writers can use as a source of mess to complicate stories, and thus make them more interesting. Also, users should be able to specify their own mess. If you want contact binaries, or accretion discs around neutron stars that are sucking the life out of a companion star, or a pair of very high speed neutron stars orbiting very close--well, you should be able to get these things and more.

Thanks again!

Gold Member
I have been doing a lot of reading lately about how the internal structures of the stars (radiative zones vs convective) and opacity are what influence the Mass-Luminosity relationship at different mass values along the main sequence, but the only information I could find on higher mass stars was L $\propto$ M.
Yes, that is for the most massive stars, that have a dominant contribution from radiation pressure rather than the gas pressure that dominates most stars. When radiation pressure dominates, it changes the way R depends on M. The radiative diffusion arguments, at roughly constant internal T, tells us that main-sequence luminosity scales roughly like R4/M. When gas pressure dominates, we find (from the virial theorem) that roughly R ~ M, so you get L ~ M3. But when M is really high and radiation pressure dominates, you reach the "Eddington limit", and internal force balance requires L ~ M, so this means the R ~ M we had for gas pressure gets replaced by R ~ M1/2, just because of the Eddington limit.
I am guessing this is because there are still lots of holes in the binaries data available for both high and low mass stars. Unfortunately, this general lack leads me to wonder how accurate the fits are. :yuck:
Yes, there is still a lot that has not been nailed down. Even simple things like metal abundances keep changing with new data, and the ability to model a star from its spectrum and get agreement with binary data has only met with partial success.
1. How close on the Mass - Luminosity and Mass - Radius relationships is 'Close Enough for Government Work?' Does anyone have any better suggestions across the range of masses?
I don't know-- it's hard to know how accurate the relationships are when we don't have access to the "raw numbers", only what we can infer indirectly. Even binaries are only a slam dunk when they are eclipsing, and even then, do we really know the metallicity of the interior? I think you should be happy with "close enough" rather than "correct."
2. How does a person like myself, trying to keep the math simple, account for age and metallicity effects on luminosity and radius? Age I have found something like a 7% increase per 100 million years, but for metallicity I haven't really found anything...
There's a surface metallicity, and a core metallicity. In low-mass stars, the surface metallicity is mixed throughout the envelope by convection, possibly even into the core for very low-mass stars. In high-mass stars, the convection is only in the core, so the core metallicity is mixed around, but differs greatly from the envelope (and fusion can occur in shells in the envelope, so you get an "onion skin" model of varying metallicity). This develops with age. So I fear the answer is, there isn't any way to treat age and metallicity that is general across the main sequence, you'd need to treat each mass range differently, or else do the full evolutionary calculation, or access tables of those who have done them. Maybe they provide fits to their tables.
3. Is there an *easy* way to estimate mass loss over time for higher mass stars? An approximation, perhaps?
Yes, there is "CAK theory", which predicts roughly that the mass-loss rate scales like luminosity to the 1.6 power, at least in any star where this would matter. The constant in the relationship is not known, because mass-loss rates are hard to measure reliably. The constant appears to work out such that a star with mass of perhaps 50 solar masses is a star with a mass-loss rate timescale equal to what its main-sequence lifetime would have been without that mass loss (so the mass loss significantly shortens the main sequence lifetime).

twofish-quant
I was of course referring to those simulations that are 3D and work through the creation of many, many stars starting with a massive cloud of gas. I don't remember how long the program I watched took to run, surely not a whole year, but I have been known to exaggerate a little to keep things light.

Different type of simulation. Anything involving 3-d modelling of fluids are extremely compute intensive and take a month to run. However, most stellar models are 1-d which means that you can nicely run them on any desktop computer.

One other funny thing. There was a paper that argued that any calculation that took more than one month to run wasn't worth running, because if it takes more than one month to run, then you save time by waiting for a faster computer, and then running it later.

Something else. Astrophysicists have to be really, really, really nice to gamers, since all of the compute power that scientists get comes from video game fanatics. You see the problem with astrophysics is that there aren't enough scientists for hardware companies to be willing to spend tens of billions of dollars developing faster comptuers. However if you have hundreds of millions of gamers that are willing to spend a few hundred or a few thousand dollars on the latest graphics card or CPU, that gives you the ton of money you need to push hardware development.

So high performance computing is really driven by the video game industry and astrophysicists get pulled along for the ride.

It looks like MESA may be just a bit more complex than what I'm going for in my own program. I'm probably going to be generating entire 'sectors' worth (hundreds at least) of stars at once, so for my purposes empirical approximations are fine (and a lot faster).

Sure that's what the professionals do. You run a massive compute which generates a table that you feed into another computer program. There's even a computational framework for doing that...

http://www.amusecode.org/

The piecewise calculations I've found and/or modified fit the stellar data I found fairly well. Do these fits for the M/L and M/R relationships approximate what we observe closely enough that those who do stellar physics would feel comfortable using them to explain the relationships of stellar properties to the layperson? (that was a mouthful) If not, is there something better?

Yes. In the end, you plot the points and draw a line.

I also can't find a chart or explanation of how EXACTLY spectral types are assigned. I know it has to do with absorption lines in the spectra, and that surface temp correlates--but how does one wind up with a F2.5V star? Is it spectra to get the letter and then effective temp to get the number(and decimal place)?

OK, what happens is that people measure the color of the star using two filters, and then from the color they assign a spectral type. Once you have a spectral type, you can infer a temperature.

One thing to remember about spectral type is that spectral types were invented way, way back in the 19th century, before people had any clue that any of this had anything to do with temperature. So what ended up happening was that people started off by saying "this is a type A star because that's what someone in 1901 said it was a type A star and it so happens that type A stars have this color which corresponds to this temperature".

To clarify the question: For stars of different starting metallicity, how do I figure out the impact of + or - metallicity on ZAMS properties?

You can run mesa or look it up on web tables,

http://stev.oapd.inaf.it/cgi-bin/cmd
http://stev.oapd.inaf.it/YZVAR/cgi-bin/form
http://stellar.dartmouth.edu/~models/isolf.html

Also if you aren't familiar with it already