Appendix A
The mass-luminosity relationship for a fully mixed, chemically
homogeneous Sun-like star was described in Chapter 3. Indeed,
Equation (3.12) indicates that the luminosity L is related to
the mass M and the chemical composition via the relationship
L = L0 75 (1 + X)
−1 M5, where L0 = L(X = X0) is a constant,
is the mean molecular weight and 0 ≤ X ≤ X0 is the hydrogen
mass fraction of the stellar gas. When the mass fraction of the
chemical elements other than hydrogen (X) and helium (Y) within
a star are small (which corresponds to the condition Z ≈ 0), the
expression for the mean molecular weight simplifies1 to (X) ≈ 2(1
+ X)
−2. With this approximation for the mean molecular weight,
the mass-luminosity relationship becomes
L X = L0 1+X −16 M5 (A.1)
Equation (A.1) determines the luminosity of a fully mixed star of
mass M and hydrogen mass fraction X.
In order to determine the effect of mass loss upon our model
star, we assume that the mass loss rate is proportional to the star’s
luminosity and accordingly write,
M
t = N
L
c2 (A.2)
where M is the amount of mass lost by the star in the time
interval t, N is a numerical parameter that can vary from zero
(indicating no mass loss) to a value as high as several hundred, L
is the luminosity, and c is the speed of light.
To make further progress Equation (A.2) needs to be converted
into an expression that varies with the hydrogen mass fraction
X, rather than time t. This, however, is easy to do. Recall from
Chapter 3 that the energy generated by the fusion reactions at the
217
218 Rejuvenating the Sun and Avoiding Other Global Catastrophes
center of a star exactly compensates for the energy lost into space
at its surface. Furthermore, to generate the energy that the star will
eventually radiate into space, a certain amount of the hydrogen
must be consumed. The amount of hydrogen consumed X in the
time interval t is determined, therefore, by the relationship
X
t = − L
QM (A.3)
where Q is the energy liberated per kilogram of stellar material by
nuclear fusion reactions, and the negative sign indicates that the
hydrogen mass fraction decreases with time.
Now, combining Equations (A.2) and (A.3) the variation in the
mass of the star can be expressed as
M
M = −N Q
c2 X (A.4)
Equation (A.4) can now be integrated to reveal how the star’s
mass changes with decreasing X. Indeed, the mass of star decreases
exponentially, and
m = exp k x −1 (A.5)
where, the following short-hand notation has been introduced
m = M X
M X = X0 x = 1+X
1+X0
(A.6)
and where the mass loss parameter is given by the expression
k = N Q
c2 1+X0 (A.7)
with X0 = 0.7 being the initial hydrogen mass fraction, and where
we have assumed that N > 0. If we also introduce the notation
l = L(X) / L(X = X0), then the luminosity equation (A.1) becomes
l = x−16m5 (A.8)
where m is given by Equation (A.5). It can now be seen that
the higher the value of N, the greater the mass-loss rate [care of
Equation (A.7)], and the lower the star’s luminosity for all values
Appendix A 219
of X. For very high mass-loss rates the luminosity can, in fact, be
driven to values less than L(X = X0).
The value of the mass-loss parameter N need not be taken as
a fixed quantity and it can certainly be allowed to vary. Indeed,
by engineering the mass-loss rate appropriately a star can be made
to evolve with a constant luminosity (this is mass-loss Scenario 1
described in Chapter 5). It can be shown2 that the condition
L(X) = L(X = X0) = constant, is achievable provided
N1 =
16
5
c2/Q
1+X = 457
1+X (A.9)
Under the assumption that Earth’s orbital radius increases in accordance
with the conservation of angular momentum, the mass-loss
rate must be adjusted so that the quantity L( X) / d2 = constant
(this is mass-loss Scenario 2 described in Chapter 5). The latter
evolutionary condition is achieved provided
N2 =
16
7
c2/Q
1+X = 327
1+X (A.10)
If the mass-loss rate is assumed to be constant throughout the
entire hydrogen-burning phase, then Equation (A.5) reveals that
the ratio of the final mass to the initial mass will be
Mf/M0 = exp -N Q/c2
X0 (A.11)
where Q / c2 = 0.007 is the energy liberated per kilogram of stellar
material by the PP chain of fusion reactions.
Notes and References
1. From Equation (3.4) the mean molecular weight becomes
(X) = 2 / (1 + 3X) when Z is assumed to be zero. However, if
one looks at the approximation a little more closely, it turns out
that with only a small error the equation for (X) simplifies to
the analytically more convenient form used in this appendix –
see M. Beech, A novel stellar model: ‘a sacrifice before the lesser
220 Rejuvenating the Sun and Avoiding Other Global Catastrophes
shrine of plausibility.’ Astrophysics and Space Science, 168,
253–261 (1990).
2. This point is discussed in, Blue stragglers as indicators of
extraterrestrial civilizations? Earth, Moon and Planets, 49,
177–186 (1990).
Appendix B:
An Accreting Black Hole Model
The accretion-driven luminosity of a black hole is derived from
Einstein’s well-known mass-energy equivalence formula E = M c2,
where E is the energy, M is the mass and c is the speed of light. If
a quantity of mass M falls into a black hole in timet, liberating
energy E, then the accretion luminosity Lacc = E / t will be
Lacc = f
M
t
c2 (B.1)
where f is an efficiency factor accounting for the conversion of
accreted material into energy, and where M / t is the amount
of material accreted into the black hole per unit time (i.e., the
accretion rate). Now, the maximum accretion rate is typically
taken to be that which produces a luminosity Lacc = LEdd, where LEdd
is the so-called Eddington luminosity. The Eddington luminosity
for an accreting black hole corresponds to the situation in which
the radiation pressure on the infalling gas exactly balances the
gravitational attraction of the black hole. For a black hole of mass
Mbh, accreting at the Eddington rate the luminosity will be
Lacc = LEdd = 4Gc
0 02 1+X Mbh (B.2)
where it is assumed that the opacity is that due to elect
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