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?
I doubt there's any such law, especially mass to any power. Jupiter has mass, so where is the luminosity?
I doubt there's any such law, especially mass to any power. Jupiter has mass, so where is the luminosity?
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
(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
There's even a simple way to understand why L ~ M^3 for high-mass stars, if you will take a short breath first.
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.
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.
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."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?
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).Interesting. Why is that? I'm guess that there is some feedback mechanism.
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.Interesting. That means that the physics of the main sequence is an opacity thing, that is independent of the fusion reactions.
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.
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?
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...
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.
Depends on how close you want things.
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.
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.
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 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 [itex]\propto[/itex] M.
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.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 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."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?
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.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...
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).3. Is there an *easy* way to estimate mass loss over time for higher mass stars? An approximation, perhaps?
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).
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)?
To clarify the question: For stars of different starting metallicity, how do I figure out the impact of + or - metallicity on ZAMS properties?
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.