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Today in Astronomy 142
Normal stars: the main
sequence
 Relationships among
luminosity, mass and
effective temperature
Stellar evolution
 Changes on the main
sequence
 Shell hydrogen fusion and
subgiants
The Pleiades (M45), a 120 Myr-old stellar cluster consisting entirely of
main-sequence stars. Photo by Robert Gendler.
19 February 2013
Astronomy 142, Spring 2013
1
Back to Live Stars
We will now learn how to scale our results on stellar
structure and luminosity to normal stars of all masses.
 We will speak in terms of scaling relations, e.g.
M2
GM 2
Pc ∝ 4
instead of Pc =
19 4
,
R
R
because in the end we will express everything in terms of
ratios to the results on the Sun, and all the constants will
cancel out, e.g.
2
4
 M
 M
 
19 February 2013
Astronomy 142, Spring 2013
4
2
  R 
 M   R  4
=
⇒ Pc Pc  
  R 
 M   R 


 
  
Pc ( M ) 19GM R
=
=
2
4
Pc 
R
19GM
2
2
The theoretical luminosity-mass relation
Escape of photon produced at star’s center: as we have seen
(29 January), in terms of the mean free path , the number of
steps N and time t are given by
N=
3R 2
N 3 R 2
t=
=
c
c
2
The average photon mean free path is  = 0.5 cm in the Sun.
How does this scale with average temperature and density?
 Very complicated (~AST 453) to show; skip to answer:
 ∝ T 3.5 ρ 2
if M < 1 M
 or so;
∝1 ρ
if M > 1 M
 or so.
(ρ = 3 M 4π R 3 )
19 February 2013
Astronomy 142, Spring 2013
3
The theoretical luminosity-mass relation (cont’d)
This tells us how luminosity scales (lecture on 29 January):
urV
(ur = radiation energy density)
L=
t
3 

 c   4
R
π
4

4
4
T
RT
σ
≈
∝



 3 
2  c

 3R  

Hydrostatic equilibrium and ideal-gas pressure support
against the star’s weight (also on 29 January) also imply
GM 2
∝ ρT ,
P∝
4
R
P GM 2 R 3
M
∝
so T ∝ ∝
4
M
R
ρ
R
19 February 2013
Astronomy 142, Spring 2013
4
The theoretical luminosity-mass relation (cont’d)
Thus, for low-mass stars ( M ≤ 1 M
 ),
3.5
3.5
T
M
R3
4
4
L ∝ RT ∝ 2 RT ∝
R
M
ρ
2
4
F
I
M
FG IJ G J RFG IJ
H K H K H RK
∝
M 5.5
R 0.5
and for higher-mass stars ( M ≥ 1 M
 ),
L ∝ RT 4 ∝
1
ρ
∝ M3
Compromise: L ∝ M
19 February 2013
4
⇒
RT 4
F
R I F MI
∝G
RG J
J
H MK H RK
3
4
L
M
=
4
L M

Astronomy 142, Spring 2013
4
 M
⇒ L
= L
 
M
 



4
5
Comparison to experiment: L vs. M
1000000
Luminosity (L)
10000
100
1
Detached binaries
0.01
L = L ( M M )
4
0.0001
0.1
Data: Malkov 1993-2007.
19 February 2013
1
10
100
Mass (M)
Astronomy 142, Spring 2013
6
The lifetimes of stars
Note how fast luminosity increases with increasing mass.
Because of this, the more massive a star, the shorter its life.
L ∝ M4
∆E
1
∝
Thus =
, or
τ
3
L
M
3
M
τ
=
τ M3
τ ≅ 1.0 × 10
19 February 2013
10
 M 
years × 

M


3
Main-sequence lifetime (years)
pp-chain fusion energy supply: ∆E ≅ 0.03 Mc 2 ∝ M
1 . 10
13
1 . 10
12
1 . 10
11
1 . 10
10
1 . 10
9
1 . 10
8
1 . 10
7
1 . 10
6
Astronomy 142, Spring 2013
0.1
1
10
Mass ( M )
100
7
The theoretical radius-mass and temperature-mass
relations
Fusion reactions comprise a sort of thermostat: temperature
in the interior of a main sequence star is only slowly
dependent upon M and R, as we saw before (again in class on
29 January). Take T therefore to be approximately constant
within a given star; then since T ∝ M R ,
R∝M ⇒ R=
R ( M M )
On one hand, L = 4πR 2σTe4 , and on the other (due to our
“compromise”), L ∝ M 4 , so
4π R 2σ Te4 ∝ M 4
M 2Te4 ∝ M 4
Te ∝ M 1 2
19 February 2013
Te M M
⇒ Te =
Astronomy 142, Spring 2013
8
Comparison to experiment: R vs. M
Radius (R)
100
10
1
Detached binaries
R = R ( M M )
0.1
0.1
Data: Malkov 1993-2007.
19 February 2013
1
10
100
Mass (M)
Astronomy 142, Spring 2013
9
Comparison to experiment: Te vs. M
50000
Detached binaries
Te = Te  M M
Temperature (K)
40000
30000
20000
10000
0
0.1
Data: Malkov 1993-2007.
19 February 2013
1
10
100
Mass (M)
Astronomy 142, Spring 2013
10
The theoretical main sequence in the luminosityeffective temperature relation (H-R diagram)
Combine this last result again with L ∝ M 4 :
L ∝ M 4 ∝ Te8
 Te
L = L 
T
 e



8
The main sequence
All of these results are in reasonable agreement with the data,
which indicates that we have included most of the important
physics in our discussions. In fact, one needs to build quite
detailed models to do better – AST 453 style, not even AST
241 style.
19 February 2013
Astronomy 142, Spring 2013
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Comparison to experiment: L vs. Te
1000000
Detached binaries
L = L ( Te Te  )
Luminosity (L)
10000
8
100
1
0.01
0.0001
10000
Data: Malkov 1993-2007.
19 February 2013
1000
Effective temperature (K)
Astronomy 142, Spring 2013
12
Stellar evolution
Terminology:
 Individuals develop; populations and species evolve.
• That is, evolutionary changes show up in the average,
not in the individual.
• The difference is important for scientists to keep in
mind, as the non-scientific public often confuses one
for the other, and can thus be misled in their ideas
about the origin of species.
 Astronomers, unfortunately, use the term “evolution”
loosely, to describe the changes of properties of individual
stars through their lives (i.e. development) as well as the
changes of stellar properties through time on the cluster
or Galactic scale (evolution proper).
19 February 2013
Astronomy 142, Spring 2013
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Mean molecular weight
=
µ
For pure ionized hydrogen:
=
µ
For pure ionized helium:
m p + me
≅ 0.5mp
2
3.97 mp + 2 me
3
≅ 1.32 mp
In general, and in terms of mass fractions: X, Y, Z = fraction
of total mass in hydrogen, helium, total of all others,
respectively, for fully ionized gas:
mp
3
1
≅ 2X + Y + Z
4
2
µ
Mean molecular weight for ionized gas with X = 0.70, Y =
0.28, Z = 0.02 (abundances found on the Solar surface):
µ = 0.62 mp
19 February 2013
Astronomy 142, Spring 2013
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Stellar evolution on the main sequence
As hydrogen burns in the
core, fusing into heavier
elements, the mean molecular
weight slowly increases. At a
given temperature the ideal
gas law says this would result
in a lower gas pressure, and
less support for the star’s
weight:
ρ kT
P=
µ
.
In the center of the Sun today,
µ = 1.17 mp .
19 February 2013
t=0
t = 4.6 Gyr
From Carroll and Ostlie,
Modern Astrophysics, 2e.
Astronomy 142, Spring 2013
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Stellar evolution on the main sequence (continued)
Therefore, as time goes on,
 the core of the star slowly
contracts and heats up.
 the radius and effective
temperature of the star
slowly increase, in
response to the new
internal temperature and
density distribution.
 the luminosity slowly and
slightly increases, in
response to the increase in
radius and effective
temperature.
19 February 2013
From Carroll and Ostlie,
Modern Astrophysics, 2e.
Astronomy 142, Spring 2013
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Aside: the Faint Young Sun paradox
 So: before about 2 Gyr ago,
the Sun should have been
faint enough that Earth’s
surface would have been
completely frozen.
 Yet there is abundant
geological evidence for
liquid water on the surface,
continuously for the past
3.8 Gyr.
This was first pointed out by
Carl Sagan and George Mullen
in 1972.
19 February 2013
Ts
L L
Te
Kasting and Catling 2003
Here,
 1− A

L
Te = 
2 
ε
16πσ r 

Astronomy 142, Spring 2013
14
17
Aside: the Faint Young Sun paradox (continued)
The reasons are not yet clear. Majority opinion (e.g. HaqqMisra et al. 2008):
 Larger atmospheric CO2 and CH4 concentrations at earlier
times led to a much larger greenhouse effect than today.
 Larger atmospheric pressures exacerbated the greenhouse
effect by broadening the wavelength ranges over which
water, CO2 and CH4 absorb.
 Later these decreased with more abundant plant life
sequestering C and producing O2 (e.g. Kasting and
Catling 2003), and H2 escaping or going into H2O.
 Alternatively, collision-induced absorption from N2-H2
might do it (Wordsworth & Pierrehumbert 2013).
19 February 2013
Astronomy 142, Spring 2013
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Aside: the Faint Young Sun paradox (continued)
Kasting 2013
19 February 2013
Astronomy 142, Spring 2013
19
Shell hydrogen burning and the subgiant phase
Eventually hydrogen is exhausted in the very center, and the
temperature is insufficient to ignite helium fusion, but is high
enough just outside the center for a shell of hydrogen fusion
to provide support for the star. Thus
 T is nearly constant in the core (isothermal helium core),
which keeps increasing in mass owing to hydrogen
depletion.
 there is increased luminosity and further expansion of the
envelope of the star.
 there is a decrease in effective temperature.
This is called the subgiant phase: the star moves off the main
sequence, upwards and to the right on the H-R diagram.
19 February 2013
Astronomy 142, Spring 2013
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Degeneracy in the isothermal core
The subgiant phase ends when the mass of the isothermal
core becomes too great for support of the star.
 Reason for a maximum in the weight that can be
supported by pressure in the core: electron degeneracy
pressure. The core is a like a white dwarf, except with
additional, external pressure.
 Maximum fraction of mass in core (Schoenberg and
Chandrasekhar, 1942):
2
µ
M isoth. core
envelope
≅ 0.37
Mtotal
µ isoth. core
F
GH
0.62 I
F
≅ 0.37 G
J
H 1.32 K
19 February 2013
I
JK
2
= 0.08 for the Sun.
Astronomy 142, Spring 2013
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