An Introduction to Modern Astrophysics · Problem 18.13

Physical Processes in the Solar System — Carroll & Ostlie

Physics — Astrophysics · nova-fireball-expansion

Problem

In this problem, you will examine the fireball expansion phase of a nova shell. Suppose that mass is ejected by a nova at a constant rate of

\dot{M}_{\rm eject}

and at a constant speed

v

. (a) Show that the density of the expanding shell at a distance

r

\rho = \dot{M}_{\rm eject}/4\pi r^2 v.

(b) Let the mean opacity,

\bar{\kappa}

, of the expanding gases be a constant. Suppose that at some time

t = 0

, the outer radius of the shell was

R

, and the radius of the photosphere, where

\tau = 2/3

, was

R_0

. Show that

\frac{1}{R} = \frac{1}{R_0} - \frac{1}{R_\infty},

where

R_\infty \equiv \frac{3\bar{\kappa}\dot{M}_{\rm eject}}{8\pi v}.

(The reason for the ``

\infty

'' subscript will soon become clear.) (c) At some later time

t

, the radius of the shell will be

R + vt

and the radius of the photosphere will be

R(t)

. Show that

\frac{1}{R + vt} = \frac{1}{R(t)} - \frac{1}{R_\infty}.

(d) Combine the results from parts (b) and (c) to write

R(t) = R_0 + \frac{vt(1 - R_0/R_\infty)^2}{1 + (vt/R_\infty)(1 - R_0/R_\infty)},

(e) Argue that terms containing

R_0/R_\infty

are very small and can be ignored, and so obtain

R(t) \simeq \frac{vt}{1 + vt/R_\infty}.

(f) Show that the fireball's photosphere initially expands linearly with time and then approaches the limiting value of

R_\infty

, in agreement with Eq. (18.30). (g) Using the data given in the text following Eq. (18.31), make a graph of

R(t)

vs.

t

for the five days after the nova explodes. The ``knee'' in the graphs marks the end of the linear expansion period; estimate when this occurs. How does this compare with the duration of the optically thick fireball phase of the nova?

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