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Unformatted text preview: Physics 160: Stellar Astrophysics Homework #7 Due Tuesday November 22nd at 5pm in SERF 340 Reading: Carroll & Ostlie chapters 14.2, 1516
Exercise [120 pts]:
(1) [25 pts] Estimate the mean molecular weights, particle number densities
and temperatures for the following environments, and then determine their
respective Jean’s masses and assess whether a selfgravitating body could
realistically form there.
(a) [5 pts] A giant molecular cloud (assume pure H2)
(b) [5 pts] The interstellar medium (assume pure ionized H)
(c) [5 pts] The intergalactic medium (assume pure ionized H)
(d) [5 pts] The Sun’s photosphere (hint: look back at prior homeworks
for estimate of n)
(e) [5 pts] The Earth’s atmosphere (assume standard temperature,
pressure, and gas composition near sea level) (2) [55 pts] Circumstellar disks
The mass density of an axisymmetric circumstellar disk is commonly
parameterized as a function of radius r and vertical scaleheight z in the
following manner: ρ(r, z ) = ρ(r, 0)e − z2
H (r ) 2 Σ(r) = Σ0 (r/R0 )α
H (r) = H0 (r/R0 )β
where H(r) is the vertical scaleheight of the disk as a function of radius, Σ(r)
is the column density of material integrated perpendicular to the disk, R0 is an arbitrary reference radius, and α, β, ρ0, Σ0 and H0 are all constants. Note
that surface density is related to the total density by:
+∞ ρ(r, z )dz Σ( r ) =
−∞ i.e., it is the integrated mass perpendicular to the disk (units of mass/area)
(a) [10 pts] Show that Σ0
ρ(r, 0) = √
π H0 r
R0 α−β (b) [10 pts] If the disk has a total mass MD, show that, for α ≠ 2: MD (α + 2)
Σ0 =
2
2π Rout R0
Rout α Rin
Rout 1− α+2 −1 Where Rin and Rout are the inner and outer radii of the disk.
(c) [10 pts] Combine these expressions to obtain a single expression for the
density of the disk as a function of r and z: ρ(r, z ) = ρ0 r
Rout α−β
− e z2
H ( r )2 and derive an expression for the constant ρ0.
(d) [10 pts] Consider a disk with total mass 1 M, Rin = 0.1 AU, Rout = 100
AU, α = 1, β = 1 and H(r) = 1 AU at r = R0 = 1 AU (this is known as a “bow
tie” disk). Evaluate ρ0 and plot the density (on a logarithmic scale) as a
function of radius in the plane of the disk (i.e., z = 0).
(e) [10 pts] Assuming that the disk material has a constant opacity of κ = 12
m2/kg, calculate the optical depth of the disk to the central star assuming it is
viewed exactly edgeon. How many magnitudes of extinction does this
correspond to?
(f) [5 pts] Is this a protoplanetary disk (optically thick) or debris disk
(optically thin)? What class of protostar is this? (Class 0, I, II or III) (3) [40 pts] Deriving the minimum stellar mass for hydrogen fusion
The equation of state for a semidegenerate object, ignoring some constants
of order 1, can be written as (in cgs units):
7 5/ 3 P = 10 ρ η2
1+η+
1+η N/m2 where ρ is measured in kg/m3 and η is the degeneracy parameter, the ratio of
Fermi energy to thermal energy for electrons in the plasma: µe
kT
2me kT
η≡
=
kTF
(3π 2 h3 )2/3 ρNA
¯ 2/ 3 ≈ 8×10−4 T
ρ2 / 3 where µe is the number of baryons per electron in an ionized gas, a number
roughly of order 1.
(a) [5 pts] What is the polytrope index (n) for this equation of state?
Assume for this question that η << 1.
(b) [10 pts] Using the results from section 10.5 in the Carroll & Ostlie, show
that the radius of this object can be written as: η2
)
R = R0 (1 + η +
1+η
and show that R0 = 2.8×10 7 M
M 1/ 3 m [HINT: you will need to use the result that for this polytrope:] ρ= 3M
= ρc /6
4π R 3 (c) [10 pts] Using your result from part (b), show that η can be written as η = 3×10 −9 Tc M
M 4/ 3 R
R0 2 (d) [10 pts] As a brown dwarf collapses, its core density and temperature
increase, so η increases. However, at some point the core temperature stops
increasing with smaller radius; i.e., dTc/dR < 0 turns to dTc/dR > 0. Using
the equations above for R (part b) and η (part c), find the critical value of η
at which this transition occurs, and show that Tc,max = 5×10 7 M
M 4/ 3 K [NOTE: this involves solving a transcendental equation of η, which is most
easily solved numerically; i.e., let a computer explore values until you get
the right one]
(e) [5 pts] The ignition temperature for hydrogen fusion is rough 3x106 K.
Using the result from part (d), estimate the minimum mass a star must have
to achieve hydrogen fusion. Your answer should be close to the more
carefully computed hydrogen burning mass limit of 0.072 M. ...
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 Fall '08
 Norman,M
 Physics, Work

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