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Physics 136b
Homework associated with Chapter 16, Supersonic Flow Available Feb 19, 2001 Due, Monday Feb 26
As usual, if any problem is trivial for you, do not do it – simply state that it is trivial and pick some other problem or make up your own. Contact me ([email protected]) if you have a question or concern about Problems 1 & 2 , which are new. There are only three problems this week.
- If you place a gaseous (Jupiter-like) planet too close to a star then it will tend to “evaporate”: A solar wind-like outflow will develop, driven by the UV flux of the star. This limits the survival time for some of the kinds of extrasolar planets recently discovered in very short period orbits.
(a) Consider a steady, ideal gas outflow from a spherical gaseous planet. Show that the relevant set of equations (energy, momentum and continuity) can be written as:
where u is the radial gas flow, ρ is the gas density, p is the pressure, γ is the ratio of specific heats, M is the planetary mass (considered to be almost entirely below the radius of interest) and Γ is the volumetric heating rate (caused by the UV flux). Steady flow means dM/dt is a constant (on a dynamical time scale, not necessarily long time scales).
(b) From these equations, derive an equation (of the standard solar wind or transonic form):
t u drd^ ( 21 u^2 + (^) c c- 1t^ p^ - GMr^ ) = C ( r )
u dudr^ = - (^) t^1 dpdr^ - r^2
GM
dt
dM (^) = - 4rt ( r ) u ( r ) r^2
( u^2 - c^2 ) (^) u^1 dudr^ = something
c^2 / ct p
from which you should derive the conditions for a transonic flow (the only solution that crosses u < c to u>c as r increases):
where subscript s means the quantity evaluated at the critical (sonic) radius rs (the value of which is implicitly defined by this equation).
(c) From the energy equation and constant mass flux, derive the approximate result
where subscript m refers to the smallest radius that the UV effectively penetrates (called optical depth unity in UV). This result neglects um^2 and pm/ρm relative to GM/rm, which is physically plausible except at very
high fluxes.
(d) At r > rm, we have Γ~ κρFUV where κ is the UV opacity and FUV is the UV flux (this assumes strong efficiency of conversion of UV flux to heating, which is often true). And we have the statement that defines rm:
Deduce that a plausible approximation is
and offer a simple physical interpretation of this result. (“Plausible” means you need to suggest why other terms might be neglected.)
(e) Find the survival time for a 10^30 g body (half Jupiter mass) of radius 1010 cm in a UV flux of 4 x10^5 erg/cm^2 .sec (corresponding to about 0. AU orbital radius). [In reality, these kinds of bodies actually get larger as mass is removed, which means they will eventually flow over the Roche lobe, defined as the region in which the planetary gravity dominates].
cs^2 = us^2 =^ GM 2 rs^ + (c^2 - 1)d^ C (^) t ur n s
dt
dM (^) = 4 r r^2 C ( r ) dr / [ rm
GM rm
rs
# + 4 (c - 1)
5 - 3c rs
GM (^) + 4
c + 1 t u d^ C r n s
]
tl dr ~ rm
3
dt
dM (^) = GM
4 r rm^3 FUV
