Download Homework #8 1. Problem 10.21 2. The Origin of the Main Sequence

Homework #8
1. Problem 10.21
2. The Origin of the Main Sequence [50 pts]
Solving the differential equations of stellar structure is beyond the scope of this class.
It would involve a nasty computer program and figuring out things like what boundary
conditions to use for the differential equations. But we can make simple approximations to these equations that allow us to figure out the scalings (proportionality)
between the mass, temperature, radius, luminosity, etc. of stars. This involves approximating each of the differential equations as an algebraic equation. E.g., every
d/dr → 1/R where R is the radius of the star. We used this technique several times
in class to estimate the central temperature of the sun.
As an example, consider the mass conservation equation
dMr
= 4πr 2 ρ.
dr
The scaling relation inferred from this equation would be
M
∝ R2 ρ
R
or
ρ∝
M
.
R3
Not surprisingly the characteristic density goes up as the mass increases or the radius
decreases. Notice that we dropped the factor of 4π. For most of this problem, we
are only interested in how the mass, density, temperature, luminosity, etc. scale with
each other and so we don’t worry about the proportionality constants. The scaling
relations derived in this way are extremely important because they do a reasonable
job of reproducing the properties of the main sequence (as you will see).
To carry out the procedure outlined above, we need to specify an equation of state,
opacity, and nuclear energy generation rate, since these show up in our equations of
stellar structure. In this problem we will use 1. the ideal gas law for the pressure
2. κ = constant for the opacity (this is not unreasonable because stellar interiors are
ionized so electron scattering is often the dominant source of scattering/absorption;
since the electron scattering cross section is independent of density and temperature,
κ = constant). 3. nuclear energy generation via the proton-proton chain, so that
ǫ ∝ ρT 5 (the book quotes ǫ ∝ ρT 4 but T 5 is a slightly better average scaling for the
wide range of temperatures considered here and it also simplifies some of the algebra).
These expressions for the pressure, opacity, and nuclear energy generation rate apply
best to stars similar to, and somewhat less massive than, the sun.1
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For higher mass stars the CNO cycle operates; for much lower mass stars the opacity is due to other
processes (e.g., photoionization) and they become fully convective; for very high mass stars radiation pressure
dominates gas pressure. One can carry out a similar analysis to this problem in all of these regimes.
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For each part below you will likely need to use the results of the previous parts and
ρ ∝ M/R3 from above. Remember that we are primarily interested in the scalings (or
proportionality) between different physical properties of stars (M, L, ...), so for the
most part you do not need to keep constants like G, k, etc. in your equations below.
a) Use hydrostatic equilibrium to obtain a relationship between the (central) temperature, mass, and radius of the star.
b) Use the radiative diffusion equation to derive a relationship between the mass and
luminosity of the star.
Stop and think. This result was derived without using any information about nuclear
reactions. This means that, for the assumptions in this problem, the mass-luminosity
relation does not depend in detail on the mechanism of energy generation. Part f)
will hopefully help you understand how this can be the case.
c) Use the nuclear energy generation equation to derive a relationship between the
mass and radius of the star.
d) Derive the relationship between the effective temperature and mass of the star and
then the luminosity and effective temperature of the star.
In class the main sequence was defined by the fact that there was a set of well defined
relations between different observed properties of stars, in particular L(M ), L(Tef f )
(the HR diagram), and R(M ). You have now derived this result (or at least a decent
approximation to it) using the equations of stellar structure!
e) Compare the scalings you find with real data (always important). Appendix G has
L, R, M , and Tef f for main sequence stars. Compare your predicted properties of an
M0 star with the real data by scaling the properties of the sun down in mass.
f) Imagine that at any fixed temperature T nuclear energy generation were less efficient
than it is by a factor of 100 (e.g., because of a smaller probability of tunneling through
the Coulomb barrier). For a star of a given mass M determine whether the luminosity,
radius, central temperature, and effective temperature would increase, decrease, or
stay the same? If you vote for one of the first two options, by how much?
Hint: Pay attention to the proportionality constant in at least one of the equations.
g) Objects that have central temperatures below ≈ 3 × 106 K do not undergo steady
fusion of hydrogen via the proton-proton chain (the temperature is too low to overcome the Coulomb barrier). Use this information to estimate the mass, luminosity,
and effective temperature of “transition” objects, those with properties at the border
between stars (steady fusion) and brown dwarfs (no steady fusion).2
h) Problem 10.10. Also, explain in few sentences why this calculation overestimates
the importance of the CNO cycle in the sun.
When you make assumptions in carrying out a calculation it is very important to
check when they are valid. In this problem we have assumed that fusion is via the
proton-proton chain instead of the CNO cycle.
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Detailed calculations give a transition at ≈ 0.07M⊙ , L ≈ 6 × 10−5 L⊙ and Tef f ≈ 1700 K.
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i) Use the result of h), the temperature dependence of the pp chain (use ǫ ∝ T 5 for
consistency with what we did above) and the CNO cycle (ǫ ∝ T 20 ), and your inferred
scaling of central temperature with stellar mass from above to determine for which
range of stellar masses the pp chain will dominate over the CNO cycle (and thus our
assumption is valid).
In this problem we have also assumed that the ideal gas law gives the dominant
source of pressure. Photons, however, contribute an additional pressure (remember
that photons have momentum just like particles so they exert a force/pressure). This
radiation pressure is given in equation 10.19 of the book.
j) Compare the gas pressure to the radiation pressure in the center of the sun using
the central temperature and density from Problem 10.9. Then use the scaling of
density and central temperature with stellar mass from above to determine for which
range of stellar masses gas pressure dominates over radiation pressure (and thus our
assumption is valid).
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