# What does this figure imply about the luminosity of pre-main-sequence stars?

It seems to show that the luminosity of a pre-main-sequence star depends primarily on the mass of the star, and that this basic relation continues to hold, to reasonable accuracy, into the main sequence. Since there is no fusion in a pre-main-sequence star, is it correct to conclude that the relationship between the mass and luminosity of a pre-main-sequence star does not hinge on fusion physics?

• Deuterium and Lithium can be burned in PMS stars, but the luminosity is still mainly determined by Kelvin-Helmholtz contraction. – AtmosphericPrisonEscape Apr 12 '18 at 9:14
• I agree the source of the energy must be gravitational contraction, but what do you mean that the luminosity is "determined by" contraction? Being the source of is not the same thing as determining. For example, one might rather say that the rate of contraction is determined by the luminosty. – Ken G Apr 12 '18 at 9:54
• (By the way, thanks to StephenG for including the figure so nicely.) – Ken G Apr 12 '18 at 9:56
• I care not about semantics, there is a great deal of physical content in the concept of "determination." Thus, "mainly determined by" is not necessarily the same things as "is the main source of." I'll give you an example: your furnace is the main source of heat leaking out of a cracked window in winter. Does your furnace determine the rate that heat leaks out through a cracked window? The isssue is not semantic, it speaks to why you insulate your house, not change your furnace, when your heating bill is high. – Ken G Apr 12 '18 at 9:59
• The last sentence of this "question" is an absurd truism. – Rob Jeffries Apr 12 '18 at 23:21

It shows that the mass of the (proto)star is the major factor determining bolometric magnitude.

The brightness of the proto-star increases a little as it contracts and becomes significantly hotter (and the radius decreases), The star moves toward the left of the diagram. As fusion in the core becomes significant the star reaches the maximum point on the curve, this stabilises the core and the star settles into the main sequence, it cools slightly and becomes slightly less bright.

• What determines the luminosity of the star prior to fusion onset? And since fusion onset does not change the luminosity significantly, should we say that what is determining the luminosity has not changed when fusion begins? – Ken G Apr 12 '18 at 9:55
• Luminosity is determined by temperature and size. As a star is forming it is getting smaller and hotter. These two processes balance each other. Fusion stops the star from collapsing further. – James K Apr 12 '18 at 14:38
• So you are saying the onset of fusion does not set the luminosity of the star in a first-brush understanding, it only stops further contraction. I agree. I'm glad there is at least one other person on here who understands this! – Ken G Apr 12 '18 at 15:03

By definition, the luminosity of a pre-main sequence star cannot possibly depend on fusion physics, since energetically-speaking, there is no significant fusion occurring!

However, if you had plotted some lower mass tracks from sources a bit more modern than 1955 (the tracks you show are out of date and incorrect, not including the Hayashi phase for example), you would see that there is predicted to be a brief plateau in luminosity (for stars/brown dwarfs less than about 0.3 solar masses) caused by deuterium burning. This has yet to be observationally tested.

The luminosity of a convective PMS star (which is how they all begin) is determined by its mass and age. To a very close approximation $$L\propto t^{-2/3}$$, with the constant of proportionality being mass dependent. Higher mass stars are more luminous at the same age, with a shallower luminosity-mass relationship than on the main sequence.

If the star becomes largely radiative, the luminosity becomes roughly independent of time (very roughly) - that luminosity is fixed by the rate at which radiative diffusion can leak the gravitational potential energy of the contracting star. The star follows the radiative horizontal (actually more diagonal for stars lower than about 2 solar masses) Henyey track in the HR diagram.

If the star remains convective, then to first order, stars follow vertical Hayashi tracks, almost independent of their mass. The luminosity is given by the rate of change of gravitational potential energy, which is in turn limited by how fast convection can transfer heat to the surface. The Hayashi track terminates when the star becomes radiative (for higher masses) or when nuclear fusion starts (at lower masses), or never (for brown dwarfs).

A set of more modern tracks is shown below. They are from the Wikipedia page on the Hayashi track, but I believe the tracks are by Palla & Stahler.

• I agree, the pre-main-sequence mass-luminosity relation cannot depend on fusion physics for just the reason you said. You might find it interesting, therefore, to cogitate on the fact that it is essentially the same as the main-sequence mass-luminosity relation. What do you conclude from that? Your other comments are false, the paper is from 1995 and intentionally does not use the Hayashi track because it is interested in radiative diffusion physics, not convection physics. (The paper is by Henyey, originator of the "Henyey track" concept that the paper focuses on.) – Ken G Apr 12 '18 at 22:18
• Check your sources on the deuterium burning, I think you are wrong there. But above all, notice when the mass-luminosity relation first starts to hold. Really, really notice that. – Ken G Apr 12 '18 at 22:21
• Yes the paper is from 1955,I misspoke, but the tracks are not so different today-- Henyey simply started them after the Hayashi track. Another thing to notice from the plots you kindly supplied is where the mass-luminosity relation does not hold-- on the lower main sequence where the plateau, where the mass-luminosity relation starts to hold for all the higher masses, does not appear. You should see that's because the radiative diffusion physics that the mass-luminosity relation relies on has not set in for those low masses. – Ken G Apr 12 '18 at 22:25
• It's explained in astro.uni-bonn.de/~nlanger/siu_web/ssescript/new/chapter8.pdf. Notice Langer does not invoke fusion physics in the explanation of the mass-luminosity relation. I'm not allowed to tell you what you should conclude from that, but I think it's already obvious. – Ken G Apr 12 '18 at 22:29
• Figure 8.4 from the paper I just cited does not find any kind of plateau when deuterium burning begins, so that's why I think you should check your sources on that. It's the dashed curves in Figure 8.5 to which I referred when I pointed out those curves contain zero fusion physics. Anyway, it should be very clear the role of radiative diffusion in the mass-luminosity relation, but I'll leave you to draw your own conclusions. – Ken G Apr 12 '18 at 22:55