[ kd ]
hwk06
Ast 4001, 2015 October 29
Homework set 6 -- Stellar interiors, homology relations, and the H-R diagram
Turn in your solutions on Thursday, Guy Fawkes Day.
_______________________________________________________________________________________
1. In stellar astrophysics we frequently use "scaling laws," "similarity relations," "homology
relations," and "proportionalities" – all these terms mean practically the same thing.
Here's a simple math exercise that formally justifies them.
As usual, M and R denote the total mass and radius of a star, and P 0 ,  0 , etc. are
physical quantities at the star's center r = 0 . Let's identify radial location with a
dimensionless variable s  r / R . The internal density distribution can be described
by a dimensionless function f ( s )   ( r ) /  0 . Obviously f ( 0 ) = 1 and
f ( 1 ) = 0 for an idealized mathematical model.
(Draw a qualitative sketch to help make this concept very definite in your mind. We
ignore the star's outer atmosphere because it has very small thickness,  r <<< R .)
We can represent pressure, temperature, etc., with other dimensionless functions f ( s ) ,
f ( s ) , etc. They're not independent of each other, because they can be calculated from
f ( s ) via the stellar-structure equations (see below).
Two stars are mutually homologous or "similar" if the same mathematical function f ( s )
applies for both of them. They may have very different masses and sizes and densities,
but their internal mass distributions "have the same shape."
Let's assume a few elementary concepts: the relation between enclosed mass and density
m ( r ) =  4 r 2  d r , hydrostatic equilibrium d P / d r = ... , the ideal gas law P =   T ,
and gravitational potential energy E G =   ( G m / r ) d m . Together these imply the
Virial Theorem and a lot of other facts. (For simplicity let's ignore radiation pressure,
which is small in most stars.)
For stars that have the same structure function f ( s ) , formally prove these relations:
(a)
0
= C 1 M / R 3  M / R 3.
(b) P 0
= ( C2 G ) M 2 / R 4
 M2/R4 .
(c) T 0
= ( C3 G /  ) M / R
 M/R .
(d) E TOT =  ( C 4 G ) M 2 / R  M 2 / R .
Here C 1 , C 2 , etc., are dimensionless constants, usually between 0.25 and 4.
Find a set of formulae that can be used to calculate C 1 and C 2 if we know f ( s ) .
_______________________________________________________________________________________
----- problem 2 is on next page -----
4001hwk06 - p2
_______________________________________________________________________________________
2. There are really at least two versions of the Hertzsprung-Russell diagram. The
basic fully-observational version is a plot of either spectral type or B  V color
vs. magnitude observed at some wavelength (often the absolute visual magnitude).
In a “physical” H-R diagram, on the other hand, the horizontal axis is converted to
log T eff , and the vertical axis is converted to either log ( L / L sun ) or, sometimes,
absolute bolometric magnitude M bol . [ Here "log" means decimal logarithm, log 10 .
Remember that M bol = 4.75  2.5 log ( L / L sun ) . ] Both transformations,
spectral type  T eff and M V  L , require calibrations based on theoretical
atmosphere models. In this problem we'll concentrate on the "physical" type of
H-R diagram. Of course a straight line on this log-log plot corresponds to a
power-law relation between L and T eff

Find a good “physical” H-R diagram of this type, based on real data. (State where
you found it.) Look at the middle part of the main sequence, with luminosities
between about 0.02 L sun and 500 L sun , or + 9 > M bol >  2 .
(a) Estimate a best-fit power-law relation for that part of the main sequence,
i.e., find the exponent  in T eff  L  . (The main sequence is noticeably
curved, but measure its average log-log slope as well as you can.)
(b) Deduce the corresponding radius-luminosity relation, R  L  . This is a
good thing to do, because radius R is much more relevant than surface
temperature when we're trying to quantify a star's internal structure.
(You may need to invoke one of the homologous-scaling relations.)
(c) Adopting a simplified mass-luminosity relation, L  M , estimate the
average dependence of central temperature T 0 as a function of mass M .
Assuming that the Sun has T 0 = 15 million K, estimate the central
temperatures of main-sequence stars with M = 0.5 M sun and 5 M sun .
Estimate their relative luminosities, too.
Three caveats!
(1) Don't confuse absolute magnitude M bol with stellar mass M .
(2) Don't confuse surface temperature T eff with central temperature T 0 .
(3) The result in part (c) is very crude, because the genuine mass-luminosity
relation gradually turns from L  M  at the lower end of this luminosity
range, to roughly M  at the upper end.
_______________________________________________________________________________________