[mesa-users] Interesting behavior at the lower edge of the main sequence

[Jon Brase]

Greetings,

I'm fairly certain I'm not the typical MESA user, in that I'm not using  
MESA as part of formal academic research, but just on my free time to  
satisfy my curiousity about stellar evolution.

One of the things I've been looking into, which I haven't found any  
detailed treatment of in the publicly available literature, is the exact  
nature of the transition from the lower end of the main sequence to the  
high-mass brown dwarves. All I've been able to discover through reading is  
that the lower mass limit for the main sequence is around 0.08 solar  
masses and that the transition is fairly sharp, without a lot of details  
beyond that. I've been exploring this transition with MESA, and have  
gotten some very interesting results.

I'll begin with a review of the dynamics of brown dwarves and main  
sequence stars as we approach the limiting mass from either side, then  
I'll describe the behavior that I've found at the limiting mass, after  
which, as I am not completely familiar with the physics involved or with  
MESA, I will have a few questions.

Early on, everything in within a few percent of the limiting mass looks  
much the same: convective throughout, approaching the electron degeneracy  
limit on denisty, a large fraction of luminosity provided by burning H  
into He3, but still dependent on gravity to make up the balance.

On the brown-dwarf side of the limiting mass, the object reaches the  
degeneracy limit, convection stalls as the gas inside the object no longer  
responds to changes in temperature with significant changes in volume, and  
the fusion rate drops off as the object is no longer able to maintain its  
internal temperature against losses through the surface by contracting  
under gravity. The fusion rate falls off so rapidly and so soon after the  
stall in convection that I thought at first that the latter was somehow  
causing the former, but it turns out that they are independent results of  
the rising degeneracy of the object that just happen to occur at around  
the same time (this will be important later).

The stall in convection begins with the formation of a radiative zone  
around the core, which then grows swiftly inward until it reaches the  
center of the star and more slowly outward ("swift" and "slow" here are  
relative terms, as objects near the transition mass form a radiative shell  
around 100 Gyr after formation). Apparently temperature gradient in the  
core is enough to drive convection despite the rising degeneracy, and, of  
course, outside of the radiative shell degeneracy is too low to choke off  
convection.

As we approach the limiting mass from below, the brown dwarf has more and  
more of its luminosity provided by fusion before it hits the degeneracy  
limit, and takes longer and longer to hit the limit.

On the main-sequence side of the limiting mass, the star reaches the ZAMS  
before it hits the degeneracy limit, then degeneracy decreases as fusion  
heats the star up, with the HR track of the main sequence forming a more  
and more acute angle with the pre-main sequence track and the time until  
the star reaches the ZAMS getting longer and longer as the mass gets  
closer to the lower limit of the main sequence.

This is where the bit about the cessation of convection and the cessation  
of fusion being separate consequences of increasing degeneracy in brown  
dwarves becomes important: stars just above the main sequence limit  
actually do see a radiative shell form, just as in brown dwarves. The  
radiative shell actually insulates the core (far from quenching it as I  
had first thought), and so for most stars in this mass range the radiative  
shell is quite transient: its formation immediately brings the star to the  
ZAMS, whereupon rising fusion output promptly reestablishes convection.  
However, in a razor thin mass range right at the main sequence limit, the  
formation of the shell does not bring the star to the ZAMS, but it does  
insulate the core enough that gravitational contraction is able to make up  
the difference between fusion luminosity and surface luminosity for a very  
long time, during which a significant amount of fusion happens. Just as  
with a brown dwarf, the core convection zone shrinks as the radiative zone  
grows inward, but as the object lingers in this state for trillions of  
years while still fusing, the shrinkage of the core causes a significant  
composition gradient to form across the radiative zone as He3:H ratios in  
the core rise.

Finally, after 7 trillion years, the star reaches the ZAMS. Unlike more  
massive red dwarves, however, reaching the ZAMS does not reinvigorate  
convection, and the convective core continues to shrink, until the star  
has a fully radiative core. Without convection, the hydrogen in the core  
is not replenished, and as it is depleted, burning moves into a shell that  
moves outward as the star ages. However, thus far, temperatures in the  
core have been just sufficient to burn hydrogen, but have remained too low  
to burn He3, resulting in a core that is mostly He3. As the hydrogen  
burning shell moves outward, temperatures in the core become high enough  
to start He3 fusion and complete the pp chain, which causes convection in  
the core to restart for a while.

As He3 is depleted in the core, convection dies down again, and He3  
burning itself moves out into a shell. Through all of this, the star  
increases gradually in luminosity, but more rapidly in temperature, so  
that its HR track takes a wider angle to its original pre-main-sequence  
track than a typical low-mass red dwarf. The track has a couple jogs in it  
near events like He3 ignition, but is fairly straight. When the star  
reaches an age of about 90 Tyr, however, the helium shell reaches  
approximately the radius at which the radiative shell originally formed,  
and, one of two things happens: either He3 fusion falters, followed by a  
general cessation of fusion altogether and a transition to a layered  
He4/He3/H white dwarf, or (actually the first case that I ran across) He3  
fusion transitions to an abrupt thermal runaway. This presumably happens  
because of the degenerate conditions and the fact that, unlike He3 burning  
in more massive stars, almost all of the hydrogen in the He3 burning shell  
has already been converted to He3, so He3 burning isn't limited by the  
rate of He3 formation. However, it is not obvious to me why this seems to  
suddenly develop after trillions of years of stable burning when the He3  
shell reaches a given radius, rather than happening with the initial  
ignition of He3 fusion in the core.

During the runaway event, the power output of the He3 burning shell jumps  
by a factor of around 1e10 from around 1e-4.7 L_sun to about 1e5.3 L_sun,  
and the star begins an ascent up what I call the "green giant branch", to  
a maximum radius of about 0.5 R_sun (the radius before the flash is about  
0.05 R_sun). I call it a "green giant" because the star grows continuously  
hotter as it grows in radius and luminosity, which would suggest the term  
"blue giant", but that already refers to something else (and the star  
never gets bluer than a K spectral type). When the star reaches the tip of  
the GGB, it has an inert, radiative core, is convective up from the region  
where the He3 flash occured up to the surface (at the beginning of the  
flash the radiative boundary is above the point where the flash takes  
place), and reaches a maximum temperature of ~4500K and maximum surface  
luminosity of 0.1 L_sun.

The star then leaves the GGB on a track slightly to the right of it. It  
contracts in several pulses, with more minor He3 flashes ending and  
briefly reversing each contraction phase. Each contraction phase drives  
convection deeper into the star, moving the burning region further inward  
toward the core as previously inert regions receive fresh He3 and H.  
Finally, the star settles down into a fully convective object at about  
1e-2 L_sun, beginning a sort of "second main sequence". Actually, the star  
only becomes fully convective if I have use_Ledoux_criterion off,  
otherwise a semiconvective region remains near the core which never fully  
mixes, but this doesn't make much difference to the final structure of the  
star; As with more typical red dwarves, convection eventually gives way to  
a radiative core as hydrogen and He3 are depleted, and the star finally  
settles into a He4 white dwarf phase, and the profile of nuclear  
abundances from core to surface is pretty much the same in the end whether  
full convection reached the core or not. Between the onset of the "second  
main sequence" and the cessation of fusion, about a trillion years  
elapses, in which time the star maintains a fairly constant luminosity,  
moving straight left on the HR plot and crossing its previous GGB track.

The transition from brown dwarf to minimal main-sequence star to  
traditional red dwarf is very sharp for a given initial composition and  
set of starting parameters, on the order of a millionth of the mass of the  
star (meaning less than one Earth mass!). As such, the variation in the  
mass at which the transition occurs varies with changes in the composition  
of the star or simulation parameters such as the number of grid points by  
an amount much greater than the width of the transition. Probing the  
dynamics of the transition is complicated by the fact that the simulation  
seems to be rather "noisy" with respect to changes in mass on the order of  
the width of the transition: masses over the transition mass can be found  
that evolve into brown dwarves in MESA, masses under the transition mass  
can be found that evolve into traditional red dwarves, and so forth. I  
believe the reason for this is quantization noise due to finite grid  
density in MESA, and this brings me to my first two questions:

First of all, given the quantization noise, how trustworthy are my  
findings so far? I seem to be getting fairly consistent results as far as  
what the minimal main sequence star looks like (in terms of structure,  
evolution, and lifetime) when I change composition and simulation  
paramaters and find the transition mass for the new setup, so I'm inclined  
to think that what I'm finding represents something that actually happens  
for stars on the edge of the main sequence, but I'm bothered by the  
noisiness of the results. Even if my findings to this point are  
trustworthy, it would be nice to be able to characterize the behavior of  
the transition region with more resolution with respect to mass than the  
quantization noise I'm presently encountering allows for (e.g, are there  
masses for which the star reaches the main sequence, but He3 burning never  
begins? In stars for which the He3 flash occurs, does it occur at  
different depths for different masses?).

Secondly, what's the best way to increase the fidelity of MESA's results  
under very small changes in mass? I assume increasing the grid resolution  
is going to be a big part of it, but is there anything else I'd need to  
do? Are there any insurmountable obstacles I'm likely to encounter (i.e,  
is there good reason to believe that MESA is not capable of delivering  
good results for such small changes in mass whatever simulation parameters  
are used?).

Beyond these first two, I have several other questions:

As I've stated, I've been using MESA to satisfy my curiosity because I  
wasn't able to find much on this subject in the publicly available  
literature. Has there been research done in this direction that just isn't  
available to the general public or is otherwise obscure, or is this  
something that just hasn't attracted a lot of interest (for example,  
because the timeframe at which lower main sequence differentiates from the  
upper brown dwarves is on the order of 10 times the current age of the  
universe, or because the transition is sharper than currently available  
simulational tools can resolve)?

I'd also like to ask what physics needs to be taken into account in a star  
such as what I described above. I assume, given the fact that the star's  
pre-main sequence and early main sequence life is characterized by the  
progressive failure of convection in the core, that mixing parameters like  
semiconvection are probably important, but not being a professional  
astrophysicist, I'm not quite sure of their importance in this scenario or  
what sane values for them would be. In particular, whereas many parameters  
for MESA have non-zero default values, which gives me an idea of what is  
normal, alpha_semiconvection and thermohaline_coeff default to 0 (which I  
take to mean "don't model these processes"), which makes determining sane  
values for them more difficult for the non-initiated.

Regarding the pgstar Power window, I assume that the reaction categories  
shown are taken from the "Category" column in  
$MESA_DIR/data/rates_data/reactions.list ? Is it safe to change category  
names in that file, or are they used for physics as well as display (for  
example, if I put he3 burning in its own category instead of in the "pp"  
category, would it affect the application of mesh_dlog_pp_dlogP_extra)? I  
ask because of the way that H->He3 burning and He3->He4 burning end up  
happening in separate regions of the star in this scenario, which would  
make it nice to be able to have them categorized separately in the Power  
window.

Finally, does my description of the evolution of a minimal main sequence  
star pass the smell test? In other words, as a layman with interest in the  
subject, I can identify the physical processes behind the various stages  
the simulated star goes through, and it all seems fairly straightforward  
 from what I know of the physics, but is there anything that makes someone  
with more expertise say "wait a minute, this process isn't modeled, and it  
would make that phenomenon not happen, which would totally change the way  
such an object would evolve", or something to that effect? The two things  
that I think have the most potential to elicit that reaction are the weird  
timing of the He3 flash (why does He 3 burning begin stably, proceed for  
trillions of years, and then suddenly turn into a thermal runaway?) and  
mass loss due to stellar winds (the transition region between classical  
red dwarves and classical brown dwarves is less than an Earth mass across.  
Is it realistic that a minimal main sequence star would lose less than an  
Earth mass over trillions of years?).

I look forward to hearing back,

-- 
Jon Brase