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Thread: Deriving and checking mass-luminosity relation

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    Deriving and checking mass-luminosity relation

    Trying to visualize the argument...
    First, imagine a star with precisely 2 times the mass of Sun.
    Further suppose that it has precisely 2 times the radius of Sun.
    And precisely the same radial distribution of density.
    This means that at any point its density is precisely 1/4 that of Sun at 1/2 the distance from centre.
    Then the gravitational acceleration should be 1/2 that of Sun at surface and any corresponding depth.
    The weight of a column of given length would be 1/8, and since the radius is 2 times bigger, the central pressure should be exactly 1/4 that of Sun.
    1/4 pressure and 1/4 density might mean the same temperature at centre, and any corresponding depth.
    Not quite.
    For one, the radiation pressures will be independent of matter density, and equal at equal temperature.
    Let us assume that this is negligible for a star like Sun, or even 4 times less dense.
    For another, the mass density may be exactly 1/4. But the number density, and therefore ideal gas pressure for equal temperature, would not be.
    Because ionization state depends not only on temperature, but density. All atoms are ionized in the centre of Sun, and light atoms like H and He are completely ionized; but heavy ions hold on to inner electrons, and there are electrons which are held by ions in centre of Sun which would be ionized at exact same temperature but 1/4 the mass density. Therefore 1/4 the mass density will mean more than 1/4 particle density and more than 1/4 particle pressure.
    Let us assume that this also is a negligible effect, though.
    Then is it correct to state that a star with 2 times the mass and 2 times the radius of Sun should have closely equal interior temperature?

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    Aren't you off the Main Sequence with 8x the volume for only 2x the mass? [Perhaps less than I thought since a 2x mass might be around a 1.6x or so R.]
    Last edited by George; 2018-May-29 at 04:17 PM.
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    Quote Originally Posted by George View Post
    Aren't you off the Main Sequence with 8x the volume for only 2x the mass?
    Yes, we are indeed off the main sequence at that point.

    Now, continue with derivation.
    A star with 2 times the radius has 4 times the area of any corresponding surface to leak energy.
    At equal temperature, suppose that the conductivity is inversely proportional to density. Which is 1/4 that of Sun.
    The radius is 2 times bigger. Therefore, the star should be 2 times worse insulator than Sun (1/4 times the density, but 2 times the thickness of insulating layer).
    With 4 times the area and 2 times poorer insulation, the star should be emitting 8 times the luminosity of Sun.
    Correct?

    Next, how much energy should the star produce?
    At equal temperature, a nucleus should have equal probability of fusing when it collides.
    But at 1/4 the density, a nucleus will collide 4 times less often.
    There are 2 times as many nuclei. So the star should produce 1/2 the energy Sun does, while losing 8 times as much.
    It should be losing energy and contracting, as a protostar on a Henyey track.

    Now have a look at the other end.
    Suppose a star has 2 times the mass of Sun, and exact same radius, and mass distribution - 2 times the density at each depth.
    Then the star has 2 times the gravitational acceleration at each depth, 4 times the weight of each gas column, and 4 times the pressure at each depth.
    Correct?
    Sticking to the assumptions that contribution of radiation pressure to total pressure is negligible and that additional ionization of inner electrons of heavy ions also is negligible, it can be said that ideal gas should have about 2 times the temperature in order to have 4 times the pressure at 2 times the density.

    What should the luminosity be?
    The radiating area is the same.
    Radiating area is the same.
    Radiation density at any depth is 16 times bigger.
    The amount of matter in the way of radiation is 2 times bigger.
    Therefore the luminosity should be 8 times bigger.

    Note the matching conclusions!
    Whether it is the same temperature and different radius or the same radius and different temperature, the total escaping luminosity ought to be 8 times that of Sun.
    Actually, it can be derived that the luminosity also ought to be 8 times that of Sun for any radius and temperature in between. Omitting the derivation now.
    Correct?
    Last edited by chornedsnorkack; 2018-May-29 at 05:38 PM.

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    So much about first principles deriving.
    But now to checking.
    There happens to be an easily observable star close to this.
    Sirius A.
    Happens to be so nearby that its parallax can be resolved.
    Happens to be a visual binary with a resolved orbit, so that its mass can be estimated independently.
    The mass is 2,06+-0,02 Solar masses. Nicely on main sequence...
    the above fundamental derivation should give 8,7+-0,2 times Solar luminosity, right?
    Wrong.
    Sirius A easily verifiably has 25 times Solar luminosity.
    What is the pretext for such a supposedly fundamental relationship to be so far wrong?

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    Quote Originally Posted by chornedsnorkack View Post
    Then is it correct to state that a star with 2 times the mass and 2 times the radius of Sun should have closely equal interior temperature?
    Yes, this can be regarded as a consequence of the "virial theorem," as the two stars have the same average gravitational potential energy per gram.

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    Quote Originally Posted by chornedsnorkack View Post
    So much about first principles deriving.
    What is the pretext for such a supposedly fundamental relationship to be so far wrong?
    Because it ignore stellar physics and basic geometry.

    https://en.wikipedia.org/wiki/Mass%E...osity_relation

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    Quote Originally Posted by chornedsnorkack View Post
    With 4 times the area and 2 times poorer insulation, the star should be emitting 8 times the luminosity of Sun.
    Correct?
    Yes, though the assumption that the opacity per gram stays fixed makes this only approximate. Still, as approximations go, it's in the right ballpark.
    At equal temperature, a nucleus should have equal probability of fusing when it collides.
    The problem here is, fusion is extremely temperature sensitive, so anything that breaks the exact equivalence between the two stars will produce a slight reconfiguring of the core temperature. Nothing else in the star cares much, but the fusion rate does.
    But at 1/4 the density, a nucleus will collide 4 times less often.
    There are 2 times as many nuclei. So the star should produce 1/2 the energy Sun does, while losing 8 times as much.
    It should be losing energy and contracting, as a protostar on a Henyey track.
    Yes, in your exact equivalence. Note also that since you have not required an energy balance, being on the Henyey track is all about the radiative diffusion rate, and fusion is doing its own thing (whether a shortfall or excess).
    Now have a look at the other end.
    Suppose a star has 2 times the mass of Sun, and exact same radius, and mass distribution - 2 times the density at each depth.
    Then the star has 2 times the gravitational acceleration at each depth, 4 times the weight of each gas column, and 4 times the pressure at each depth.
    Correct?
    Yes.
    Whether it is the same temperature and different radius or the same radius and different temperature, the total escaping luminosity ought to be 8 times that of Sun.
    Actually, it can be derived that the luminosity also ought to be 8 times that of Sun for any radius and temperature in between. Omitting the derivation now.
    Correct?
    Yes, that is the heart of the "Henyey track," and also the mass-luminosity relation for main-sequence stars that fall on the Henyey track. You have officially understood why the mass-luminosity relation has little to do with fusion-- but don't expect the intro textbooks to be on that page!

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    Quote Originally Posted by chornedsnorkack View Post
    Sirius A easily verifiably has 25 times Solar luminosity.
    What is the pretext for such a supposedly fundamental relationship to be so far wrong?
    It's because the opacity per gram is quite sensitive to internal temperature in that mass range. The usual mass-luminosity relation works approximately over three orders of magnitude in the mass, but it doesn't work over that particular factor of 2, because that's just where the opacity is changing a lot. Low-mass stars also give deviations because they are so convective, rather than radiative, and they can even start to exhibit the influence of degeneracy as you get down toward the brown dwarfs. So as with so many things in astronomy, you have to choose your poison when you start making approximations-- you kind of have to gloss your eyes and see the big picture rather than the details.
    Last edited by Ken G; 2018-May-29 at 07:30 PM.

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    Ah, I see.
    So the standard derivation is grossly wrong, and it is the assumption of "opacity dependent on mass alone" that is far from truth.

    Can it be said that Sun is abnormally dim, because the interior of Sun is abnormally good insulator?

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    Quote Originally Posted by chornedsnorkack View Post
    Ah, I see.
    So the standard derivation is grossly wrong, and it is the assumption of "opacity dependent on mass alone" that is far from truth.
    I'm not sure I'd say grossly wrong, perhaps significantly wrong but also significantly right. Especially for more massive stars that mostly have free-electron opacity, which is indeed constant. But there is always a tradeoff between complexity and accuracy, so it's not right or wrong to choose a particular compromise there, it is only important to understand what compromises are being made. It's often a subjective judgement about what is or is not a good tradeoff there, and it certainly depends on the goals for the enterprise. For example, if you are trying to get someone to understand stars, it may be all right to simplify the opacity physics, because replacing them by something more realistic changes the quantitative agreement but not really the general idea. What one must avoid is getting the physics completely wrong, like using fusion rates instead of radiative diffusion rates, or not noticing when convection takes over as the key process.
    Can it be said that Sun is abnormally dim, because the interior of Sun is abnormally good insulator?
    I think that's true if you are comparing to much more massive stars, though perhaps not if you are comparing to lower mass stars. A good way to frame the situation might be to start with very massive stars (though not too massive, if you get near the Eddington limit you get important radiation pressure and that changes the physics) and assume the constant opacity per gram of pure free electrons. That gives your result. Then as you lower the mass, recognize that interior temperatures will drop, and more and more electrons will be bound, which tends to increase the opacity. This reduces the luminosity below your assumption, causing the luminosity to fall more steeply as mass is lowered. The steeper fall makes the mass-luminosity exponent more like -3.5 on average, rather than the simpler -3 you got. But even the -3.5 is just an average increase in steepness; in the vicinity of the Sun it's more like -5. So rather than saying the Sun is abnormally dim, we can say that it lives in a region where the opacity is rising abnormally steeply. It's the drop in L with M that is abnormal, the L itself depends on what you compare it to.
    Last edited by Ken G; 2018-Jun-01 at 02:26 PM.

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    Quote Originally Posted by Ken G View Post
    I think that's true if you are comparing to much more massive stars, though perhaps not if you are comparing to lower mass stars. A good way to frame the situation might be to start with very massive stars (though not too massive, if you get near the Eddington limit you get important radiation pressure and that changes the physics) and assume the constant opacity per gram of pure free electrons. That gives your result. Then as you lower the mass, recognize that interior temperatures will drop, and more and more electrons will be bound, which tends to increase the opacity. This reduces the luminosity below your assumption, causing the luminosity to fall more steeply as mass is lowered.
    The cause is not directly mass, though. It is temperature.
    Does it mean that a Sirius-like protostar whose interior temperature matches that of Sun would have luminosity as predicted by the Eddington derivation - less than 9 times that of Sun - and as it contracts and heats up, rather than doing so at constant luminosity as follows from the assumption of constant opacity, should brighten up on Henyey track?

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    Quote Originally Posted by chornedsnorkack View Post
    The cause is not directly mass, though. It is temperature.
    Actually, the temperature doesn't matter, because the only way to get to high temperature is to constrict the star. The higher radiative energy density compensates for the lower volume. So that's why the Henyey tracks are at constant luminosity -- neither temperature nor radius matters, only mass (as long as you assume constant opacity per gram).
    Does it mean that a Sirius-like protostar whose interior temperature matches that of Sun would have luminosity as predicted by the Eddington derivation - less than 9 times that of Sun - and as it contracts and heats up, rather than doing so at constant luminosity as follows from the assumption of constant opacity, should brighten up on Henyey track?
    Including more accurate opacity doesn't seem to spoil the constant luminosity, it only changes what the luminosity is. So it seems to work out that even when you include a more detailed opacity, temperature and radius still don't matter, only mass does. That's the fundamental reason the "mass-luminosity relation" also holds in the pre-main-sequence. Lots of places will tell you it is a rule for the main sequence only, but that's mostly wrong, because it's very important that the mass-luminosity relation holds on the pre-main-sequence-- it's how you derive the mass-luminosity relation, you shouldn't need fusion physics. The only time you need fusion physics is when you never reach the Henyey track, for the very low mass stars, but the mass-luminosity relation should be regarded as different for them anyway.

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    It would be interesting to project how stars of various masses would progress if thermonuclear fusion never occurred. As Ken G mentioned in a thread about 12 or 13 years ago, Eddington and his colleagues calculated the mass/luminosity relationship to be expected for for main sequence stars before anyone had envisioned fusion, and they came close to the observed values. This was on the basis of what was already known about the thermodynamics of hot gas, including the heat transfer rate at different densities. Eddington predicted that a protostar's contraction would slow almost to a halt as it reached a critical density for its mass, and then would be in a quasi-steady state on the main sequence. We now know that as it eases into this state it induces just enough fusion to keep it hot for billions of years in the case of the Sun. I will now stick my neck out with an educated guess that in the absence of fusion it would continue a very slow contraction along an extension of the Henyey track for a few millions or tens of millions of years and then start diminishing in luminosity as it continues to contract toward becoming a white dwarf. This is similar to what is predicted for a minimum mass red dwarf as described in Sky and Telescope in the November 1997 issue. Those stars are convective throughout and never become stratified in the way that leads to a red giant stage as they evolve. A more massive star would evolve much faster, and if massive enough, undergo a core collapse that may or may not make a supernova but definitely end up with a neutron star or a black hole.

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    Quote Originally Posted by Ken G View Post
    Actually, the temperature doesn't matter, because the only way to get to high temperature is to constrict the star.
    But constricting the star takes time on Henyey track. Temperature is not equal for same mass stars on different places along Henyey track.
    Quote Originally Posted by Ken G View Post
    The higher radiative energy density compensates for the lower volume. So that's why the Henyey tracks are at constant luminosity -- neither temperature nor radius matters, only mass (as long as you assume constant opacity per gram).
    But the derivation of constant luminosity on Henyey track depends on the assumption of constant opacity per mass.
    Assumption shown to be wrong.
    Does it follow that if opacity per mass changes on Henyey track, so does luminosity?
    Quote Originally Posted by Ken G View Post
    Including more accurate opacity doesn't seem to spoil the constant luminosity, it only changes what the luminosity is. So it seems to work out that even when you include a more detailed opacity, temperature and radius still don't matter, only mass does. That's the fundamental reason the "mass-luminosity relation" also holds in the pre-main-sequence. Lots of places will tell you it is a rule for the main sequence only, but that's mostly wrong, because it's very important that the mass-luminosity relation holds on the pre-main-sequence-- it's how you derive the mass-luminosity relation, you shouldn't need fusion physics.

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    Quote Originally Posted by Hornblower View Post
    I will now stick my neck out with an educated guess that in the absence of fusion it would continue a very slow contraction along an extension of the Henyey track for a few millions or tens of millions of years and then start diminishing in luminosity as it continues to contract toward becoming a white dwarf.
    Yup, that's it, except for the highest mass stars which would undergo core collapse before becoming what dwarfs, as you point out.

    A detail to add is, as solar-like stars got hotter in this hypothetical scenario, they would start to have free electron opacity dominate, which will actually reduce the interior opacity so the luminosity will eventually rise up a little on the path to a white dwarf, but that kind of detail we can leave out. (You can see the rising luminosity effect in figure 2 of https://www.researchgate.net/figure/...l_fig1_1792475 where it is too small to matter at the same level of approximation where fusion physics is being neglected. But if fusion never happened, the opacity drop might eventually start to be an important effect.)
    This is similar to what is predicted for a minimum mass red dwarf as described in Sky and Telescope in the November 1997 issue. Those stars are convective throughout and never become stratified in the way that leads to a red giant stage as they evolve.
    Yes, no red giants because because the red giant effect fundamentally requires shell fusion (regardless of if there is convection). I don't know what it says in Sky and Telescope, but if there were no fusion, very low-mass stars should eventually find their Henyey tracks and make a sharp turn in the H-R diagram. So they would continue to keep a cool surface temperature and drop in luminosity as they contract, until convection starts to get replaced by radiative diffusion, at which point they would keep their (low) luminosity and start evolving to the left as their surface temperature rises and their radius contracts, eventually becoming white dwarfs. If they were very very low mass, they would become white dwarfs before ever reaching their Henyey tracks, and basically be large gas giant planets.
    Last edited by Ken G; 2018-Jun-03 at 02:14 PM.

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    Quote Originally Posted by chornedsnorkack View Post
    But constricting the star takes time on Henyey track. Temperature is not equal for same mass stars on different places along Henyey track.
    Exactly, that's how we can see that temperature doesn't really matter in the mass-luminosity relation.
    But the derivation of constant luminosity on Henyey track depends on the assumption of constant opacity per mass.
    Assumption shown to be wrong.
    Yes, but not badly wrong. It's at the level of a "first order understanding", where you start with the crucial effects, and then layer in the details as your need for accuracy increases. At some point in that process, you just throw in the towel and do a "black box" simulation that includes all the physics but yields very little of conceptual value. The important thing is not to do that first, or else you will end up with what many authors of intro textbooks do, where they try to apply faulty intuition after the fact when they already know how the simulations come out, and don't seem to care if the faulty intuition makes sense or not because they know how the answer comes out.
    Does it follow that if opacity per mass changes on Henyey track, so does luminosity?
    Yes, there is a small increase in luminosity as the temperature rises and bound electrons are lost, see e.g. figure 2 in https://www.researchgate.net/figure/...l_fig1_1792475 .
    Last edited by Ken G; 2018-Jun-03 at 02:13 PM.

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