Download Homework #2 1. There are two ways to estimate the energy carried

Homework #2
1. There are two ways to estimate the energy carried by convection. The first is that the energy
flux is Fc ≈ 1/2ρvc3 ≡ Fc,1 where vc is the characteristic velocity of the convective motions.
This is the KE flux carried by moving blobs. The other estimate is that Fc ≈ ρ∆Evc ≡ Fc,2
where ∆E is the difference in the thermal energy of a rising hot blob (or sinking cool blob)
relative to the background star (where E is per unit mass). Show that these two expressions
are equivalent, to order of magnitude (which is the accuracy of mixing length theory).
2. The solar convection zone contains very little mass (only ≈ 2% of the mass of the sun).
Consider a model in which we neglect the mass of the convection zone in comparison to the
rest of the sun. Model the convection zone as a polytrope with P = Kργ . The radius of the
base of the convection zone is Rc .
a) Solve for the density, temperature, and pressure as a function of radius in the convection
zone. Do not assume that the convection zone is thin (i.e., even though Mr = constant = M ,
because r changes significantly in the convection zone, do not assume that the gravitational
acceleration is constant).
b) In detailed solar models, the pressure at the base of the convection zone is ≈ 5.2 × 1013
dyne/cm2 and the density is ρ ≈ 0.175 g cm−3 . Estimate the radius of the base of the
convection zone Rc . Compare this to the correct answer of Rc ≈ 0.71R .
c) In your model, what is the temperature of the sun at 0.99R , 0.9R , and at the base of
the solar convection zone. This gives you a good sense of how quickly the temperature rises
from its surface value of ≈ 5800 K as one enters the interior of the sun.
d) Using a simple publically available stellar structure code, I find that the zero-age main
sequence (ZAMS) solar model has the following properties (ZAMS is when the star first starts
to undergo fusion, so no H has yet been converted into He; this is t = 0 in stellar models, vs.
t ≈ 4.5 billion years for the sun today).
log(Tef f ) log(L/L ) log(Tc ) log(ρc ) log(R/R )
3.7256
-0.3287
7.1287
1.8444
-0.0886
For the same code, if I turn off convection, so that energy is always transported by radiation
even when the solution is in reality convectively unstable, I find the following solar model,1
log(Tef f ) log(L/L ) log(Tc ) log(ρc ) log(R/R )
3.6978
-0.3292
7.1286
1.8443
-0.033
The luminosity, central temperature, and central density do not change significantly (not
surprising since the convection zone has so little mass). However, the radius of the model sun
increases by about 14% and the effective temperature decreases as a result by about 7%.
To understand this, consider the structure of a radiative atmosphere in which gas pressure
dominates and the enclosed mass, enclosed luminosity and opacity are all constant (constant
Mr and Lr are quite reasonable at large radii, while constant κ is only approximately valid
and is made for simplicity). Use hydrostatic equilibrium and the radiative diffusion equation
to derive a relationship betweeen pressure and density in such an atmosphere. Use this and
your model atmosphere from a) to explain the numerical results above, namely that the radius
of the sun would be larger if the outer convection zone were absent. Note: You showed in
HW 1 that an n = 3 polytrope is a pretty good model of the radiative parts of the sun. This
problem should help you see why this is a reasonable first approximation.
1 This
was not done with MESA. We had trouble doing this in MESA. Extra credit if you can do so.
1
3. In class, we discussed Kelvin-Helmholz (KH) contraction of fully convective stars to the main
sequence.
a) Argue that, for KH contraction to occur, the timescale for KH
p contraction tKH must be
longer than the gravitational free-fall time of the cloud, tf f ≈ 1/ Ghρi, where hρi is the mean
density of the cloud. What happens if tKH < tf f ?
b) Estimate the critical radius Rc (in R ) at which tKH ≈ tf f , i.e, at which KH contraction
begins, for a given cloud of mass M (in M ). Assume, as we did in class, that the cloud is
fully convective at early times. Show that for R < Rc , the cloud undergoes KH contraction
according to your criterion from a). Recall that the luminosity of a fully convective star is
L ≈ 0.2L (M/M )4/7 (R/R )2 .
In lecture we showed that moderately massive stars become radiative well before they reach
the main sequence.
c) Does the transition from convective to radiative first happen at the center of the star
or the outside (i.e., is the transition inside-out or outside-in)? To determine this, consider
where the star is likely to first become stable by the criterion d ln T /d ln P |rad < 2/5. During
KH contraction the energy source for the star (gravity) depends only weakly on density and
temperature so assume that Lr /Mr ∼ constant, independent of position in the star.
In the rest of this problem, we will derive the KH contraction of a radiative star. Assume for
simplicity that electron scattering dominates the opacity of the star.
d) Derive R(t,M) for radiative stars undergoing KH contraction.
e) If stars reach the main sequence when R/R ≈ M/M , what is the time to reach the main
sequence, tM S , as a function of stellar mass?
4. The treatment of convection, even within the simplified framework of mixing length theory,
has numerous complications. In this problem, we will use MESA to explore a few of them.
We will consider the main sequence evolution (core hydrogen burning) of a 3 M star. We
will adjust some of the convective parameters used by MESA and assess their impact on the
mass of the convective core (as a function of time) and the main sequence lifetime of the star.
So that you don’t have to run a dozen MESA models yourself, each student will be assigned a
model to run. Complete your assigned model by Wed Feb 20 and send it to Josiah, who will
collect them and distribute a full set to everyone for the interpretation parts of the problem.
For each part below, use qualitative physical arguments to explain the variation found in the
MESA calculations in the size of the convective core and main sequence lifetime with the
relevant convective parameter.
a) When using mixing length theory, one must choose the mixing length parameter α. Explore
the effects of varying α; explain physically what you find.
b) A rising (or falling) blob will not instantaneously halt when it reaches the edge of the region
of convective instability. Its inertia will cause it to penetrate some distance into the stable
region. This convective overshooting introduces additional mixing.2 Explore the effects of
varying the overshooting parameter fov ; explain physically what you find.
c) The Schwarzchild criterion for convection (which is what we derived in lecture) omits the
possible effects of composition gradients; the Ledoux criterion, which we will develop in this
problem, takes them into account.
2 In MESA, the overshooting parameter f
ov is the length scale (in units of one-half the pressure scale height) for
the exponential decay of the convective diffusion coefficient.
2
i) Consider an ideal gas. Relate the logarithmic density change δ ln ρ = δρ/ρ to the logarithmic
changes in P, T, µ. The condition for whether a blob will be convectively unstable can be
expressed (as in class) in terms of the Brunt-Väisälä frequency via the density contrast
N2 = g
(ρblob − ρbackground )
<0
ρbackground δr
In addition to the assumptions of adiabaticity and pressure equilibrium from class, assume
that the blob does not mix with the background medium to simplify your expression for N 2
and write it in terms of background gradients.
ii) What is the sign of (d ln µ/dr)background that you expect to develop during core H burning?
Given your result from (i), will the composition gradient have a stabilizing or destabilizing
effect on convection?
iii) Explore the effect of using the Ledoux convective criterion instead of the Schwarzchild
criterion; explain physically what you find.
d) Regions that are stable by the Ledoux criterion, but unstable by the Schwarzchild criterion
are referred to as semiconvective. While dynamically stable, instabilities can grow on longer
timescales in these regions and cause some mixing to occur.3 The exact efficiency of mixing by
semiconvection is uncertain and needs to be calculated using multi-dimensional simulations.
It is typically modeled with a parameterized efficiency αsc analogous to the mixing length α
in part a).4 Explore the effect of varying αsc ; explain physically what you find.
3 Semiconvective regions are overstable on the thermal diffusion timescale. A blob which is perturbed will oscillate
around its equilibrium position. But as the blob rises, it is not perfectly adiabatic and so loses some heat. When it
returns to its old equilibrium position, it is now cooler than the surroundings and so proceeds to sink deeper, where
it is heated and when it returns to the old equilibrium is now hotter than the surroundings and so rises to an even
higher position, etc.
4 In MESA, the diffusion coefficient associated with mixing of the composition by semiconvection scales linearly
with αsc .
3