Problem Set 3

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Physics 688, From Brown Dwarfs to Giant Planets

Stanimir Metchev, Mar 16, 2009

Problem Set 3

Due in class on Monday Mar 23, 2009

Each letter part of a problem is worth 5 points, unless otherwise noted. Material covered: lectures 13–20.

1.

(10 points)

Problem 4 in Chapter 3 of Reid & Hawley.

2.

a)

(10 points)

Using the equations for hydrostatic equilibrium and mass continuity, show that the gravitational potential energy of a polytrope of index

n

(i.e,

P =

K

ρ

1+1/

n

) is given by

" # $

M

%

0

GM r r dM r

= $

3

5

$

n

GM

R

2

.

Hint: Show that

dP

"

=

K n

+

n

1

"

1

n

#

1

d

" =

(n

+

1)d

$

&

%

P

"

'

)

(

.

You’ll also need to use

!

integration by parts several times. b) Then prove equation (2.45) in Burrows & Liebert (1993, RvMP, 65, 301). c) Derive equation (2.46) in Burrows & Liebert (1993), which gives an approximate

!

Note

: My derivation produces an

η

(

α

+

η

)

–2 dependence, rather than a (

α

+

η

)

–1 dependence on the right hand side of (2.46) in Burrows & Liebert (1993). I will accept either result as a correct answer, but would be happy to be proven wrong.

3.

a)

(10 points)

Starting from equations (2.26) and (2.30) in Burrows & Liebert (1993), and using the expressions for the enhancement of the nuclear fusion rates due to electron screening of the nuclei (2.28–2.29), show that the temperature and density dependencies of the energy generation rate in the interior of a minimum mainsequence mass (MMSM) star have power law indices of 6.31 and 1.28, respectively. Assume central temperature of

T c

= 3

×

10

6

K and central density of

ρ

c

= 10

3

gm/cm

3

for the MMSM star. b)

(10 points)

Using equation (2.27) in Burrows & Liebert (1993) and appropriate estimates for

T c

and

ρ

c

(e.g., from Burrows et al. 2001, RvMP, 73, 319, or from equations 2.33, 2.38, and 2.40 in Burrows & Liebert [1993] for a

X

= 0.75,

Y =

0.25 composition), derive the temperature and density dependence of the energy generation rate in the core of a brown dwarf at the deuterium burning mass limit.

4.

(10 points)

Explain the role of condensate clouds and dust in determining the 0.5–15 micron spectral energy distributions of L dwarfs. Why is an L dwarf with a cloudy atmosphere expected to be redder in the near infrared (1–5 micron) than a hypothetical L dwarf of the same effective temperature with a clear atmosphere of the same effective temperature? What would the effect on the level of the cloud deck and on the near-infrared color of the L dwarf be under lower gravity (e.g., as in a very young giant planet)?

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