Sec on 2.
Stellar structure and proper es (I)2.1 Hydrosta c equilibrium

tititi

2.2 Nuclear burning

Let’s understand these ques ons
with the words of physics• Why are stars so luminous?
• Why do stars show L ~ M ?
• Why do stars evolve?
• Why does the des ny of stars depend on the mass?
• Why do some stars explode?
• Why don’t normal star explode?
• Why does stellar core collapses?
• Why is the energy of supernova so huge?
•…
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4

Our sunL = 4 x 1033 erg/s = 4 x 1026 J/s (W)Electronic power consump on in Japan
1.5 x 1019 J / year==> Japanese power consump on for 2 x 107 yr= solar radia on in 1 second

HNuclearburning

Hetititi

JAXA/ISAS

Energy source: E = mc2

ヘリウム

0.7% heavier
水素4 x mp

水素原子核 4 つ＞

ヘリウム原子核➡

1kg のヘリウム0.7％重い

1.007 kg の水素mαQ1: How much energy is released for one nucleon? (核子あたり)2
E ＝ mc2 ＝ 0.007 kg ×（3×10 8 m/s）

E = Δmc2＝ 6×1014 ジュール

Solar mass：2 x 1033 g

Q2: How much energy does the Sun can produce?
(assume 10% of solar mass can be used for nuclear burning)
Q3: How many years the sun can keep shining?
L = 4 x 1033 erg/s

1 yr ~ 3 x 107 sec

Binding energy of nuclei
Eb = [NmN + Zmp - mi] c2
P+nNuclei

Higher binding energy= strongly “bound”= more stable= “lighter”

Fe is the most stable nucleus

What is going on at the center of the star?
How does nuclear burning occurs?

The energy levels of the compound nucleus play a crucial rolesection, see Fig. 6.2. Let Emin be the minimum energy required to r
state of C to infinity, analogous to the ionization energy of an atom
spond to bound states in an atom; these can only decay by γ-emissi
These ‘stationary’energy levels have long lifetimes τ and corre
6according to Heisenberg’s uncertainty relation

Nuclear burning
Coulomb barrier E ~ (Z1Z2e2)/r ~ 10 eV (MeV)
Typical energy of the gas E ~ kT ~ 10!3 eV (keV) <= 107 K
Γ==> Tunnel e ectsECτ.

MeVV(r)E’EenergykeV0rcrn

E min−V0Figure 6.2. Schematic depiction of thetential, shown as a thick line. The potension at distances r > rn , and by nuclea
particle with kinetic energy E at infinity
rc . The horizontal lines for 0 < r < rn innucleus formed during the reaction. Th
quasi-stationary levels with E > 0 are btimes. If the incoming particles have en
they can find a resonance in the compou~10-13 cm (1 fm)
Textbook by Pols

ffr

Condi on of H-burning

FusionreactorSi−bO−bC−b(K)log Tc~108 K

10He−b8
H−b1006101.0
0.1-104−505log ρc

10Figure 8.3. The samin Fig. 8.2, together w
the log T c – log ρc plaoccur.

(g cm-3)ti

Lecture Note by Pols

Hertzsprung-Russel diagram
Luminosity
Red giantsMainsequence

White dwarfs

Temperature (K)tth p://astronomy.nmsu.edu/geas/lectures/lecture23/slide04.html

Mass - radius rela on for the main sequencelog (R / Rsun)

1.0

Contamina on of
evolved stars

0.5

R ~ M0.70.0

Outcome of
the central propertyof the star

−0.5−1.0−101log (M / Msun)
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ius (right) relations for components of double-lined eclipsing

Lecture Note by Pols

Summary: Stellar structure and proper es (I)• Energy source of the stars• Nuclear burning• E = mc
• Stellar structure• Hydrosta c Equilibrium• Central temperature of the stars T ~ 10 K
• Require tunnel e ects for nuclear burning• Stellar proper es• Almost constant central T => R ~ M• Observed mass-radius rela on (R ~ M )27titifftiti

0.7Appendix

Virial theorem(galaxy clusters)

1993ApJ...404...38G

Velocity of galaxies in Virgo Cluster

Girardi+1993

Virial theorem(galaxies)

M-sigma rela on
The Astrophysical Journal, 764:184 (14pp), 2013 February 20

McConnell & Ma

Figure 1. M• –σ relation for our full sample of 72 galaxies listed in Table 3 and at http://blackhole.berkeley.edu. Brightest cluster galaxies (BCGs) that are also thecentral galaxies of their clusters are plotted in green, other elliptical and S0 galaxies are plotted in red, and late-type spiral galaxies are plotted in blue. NGC 1316 isthe most luminous galaxy in the Fornax cluster, but it lies at the cluster outskirts; the green symbol here labels the central galaxy NGC 1399. M87 lies near the centerof the Virgo cluster, whereas NGC 4472 (M49) lies ∼1 Mpc to the south. The black hole masses are measured using the dynamics of masers (triangles), stars (stars), orgas (circles). Error bars indicate 68% confidence intervals. For most of the maser galaxies, the error bars in M• are smaller than the plotted symbol. The black dotted linetiMcConnell & Ma 2013would be difficult to find an exact stellar match to the[Mg/Fe] ratio of any given galaxy spectrum, and it would

Velocity dispersion

Fig. 2.—Same as Fig. 1, but for the blue side (5040–5430 Å). The Mg i b
triplet and [N i] !5200 regions (dotted line) are excluded from the fit.

described above andthe blue side, the sim
observed with the 12
For both the blue
is very successful atsion over the entire r
measured dispersion
sion. The red-side mthe input dispersion
the rms disagreeme
velocity dispersionss"1 for the blue side
test of the method; athe measurement rosion, S/N, and degr
red and blue spectra
strate the accuracy oinput model. As exp
velocity dispersions
sion. On the blue sids"1, the measuremen
red side, with an in
returned # ¼ 25 # 3s"1, but for input dis
to yield useful results
We investigated oroutine, the order oadjust the template’s
Our final measurem
polynomial on bothstrates how these reBarth+2002
mial is used. The r

Central temperature of stars

Mass - luminosity rela on of the main sequence stars
6Star with M = 10 Msun=>1.0L ~ 104 Lsun=> Life me~ 1/103 of the Sun
0.5~ 1010 yr (100億年)/1034log (R / Rsun)log (L / Lsun)

L ~ M420~ 107 yr (1000万年)

0.0−0.5

More massive stars
have shorter life me−2−10log (M / Msun)

1−1.0−10log (M / Msun)

tititi

Lectureby Pols of doubleFigure 1.3. Mass-luminosity (left) and mass-radius (right) relations
forNotecomponents

imes that of a typical reaction involving only strongTheinteractions.
Therate cannotbnet resultofreactionhydrogenburning
ured in the laboratory and is only known from theory.
1 another4 proton to +from 3 He.
After some deuterium is produced, it rapidly reacts with4 H → He + 2 e + 2 ν .
quently three different branches are possible to complete the chain towards 4 He:

1a. H-burning (pp chain)

You may verify using Sec. 6.1.1 th
1 H + 1 H → 2 H + e+ + νnucleus two protons have to be con
2 H + 1 H → 3 He + γ
interactions (p → n + e+ + ν), whic!#!
##
not to include the neutrino energi!
##"!
!
$theenergy
thatisusedtoheat
the3 He + 3 He → 4 He + 2 1 H
3 He + 4 He → 7 Be + γ(including
the γ-rays resulting fro#!
!
#!
#energies
oftheresultingnuclei.
T
#!
pp1
"!
$
#
smallerthan
26.734MeV
anddep7−7
7
1
8Be + e → Li + ν

7 Li + 1 H → 4 He + 4 He

pp2

Be + H → B + γ

8 B → 8 Be + e+ + ν

8 Be → 4 He + 4 He

pp3The pp1 branch requires two 3 He nuclei, so the first two reactions in the chain have to takeTextbookby Pols
wice. The alternative pp2 and pp3 branches
requireonly one 3 He nucleus and an already existi

Energy produc on rate(per gram)q ~ ρT4Since a simultaneous reaction between four protons is extremely unlikely, a chain of reactions isways necessary for hydrogen burning. This can take place in two distinct ways: either direct fusion
protons via the p-p chain, or by using already present CNO-nuclei as catalysts in the CNO cycle.
ydrogen burning in stars takes place at temperatures ranging between 8 × 106 K and 5.0 × 107 K,
pending on stellar mass and evolution stage.

he p-p chains

he first reaction is the so-called p-p reaction:
1H + 1 H → 2 H + e+ + νorp + p → D + e+ + ν .

(6.45)

T ~ 4 x 106 K

his involves the simultaneous β-decay of one of the protons during the strong nuclear interaction.
his is very unlikely and the p-p reaction therefore has an extremely small cross-section, about 10−20
mes that of a typical reaction involving only strong interactions. The reaction rate cannot be meared in the laboratory and is only known from theory.
After some deuterium is produced, it rapidly reacts with another proton to from 3 He. Subseuently three different branches are possible to complete the chain towards 4 He:
1 H + 1 H → 2 H + e+ + ν
2 H + 1 H → 3 He + γ!!
"!#
##!!

3 He + 3 He → 4 He + 2 1 H##$3 He + 4 He → 7 Be + γ!
"!
!

pp1! !####$#

7 Be + e− → 7 Li + ν

7 Be + 1 H → 8 B + γ

pp28 Be → 4 He + 4 He

7 Li + 1 H → 4 He + 4 He

8 B → 8 Be + e+ + ν

pp3(6.46)

he pp1 branch requires two 3 He nuclei, so the first two reactions in the chain have to take placewice. The alternative pp2 and pp3 branches require only one 3 He nucleus and an already existing 4 Heucleus (either present primordially, or produced previously by hydrogen burning). The resulting 7 Beti

Textbook by Pols

Textbook by Prialnik

The CNO cycle

16E
produconrateq~
ρT1b.
Hburning(CNOcycle)
If some C, N, and O is already present in the gas out of which a star forms, and if the temperatu

7 Kcycle. This is a cyclic
is sufficiently high, hydrogen fusion can take place via theCNO
T ~so-called
1.5 x 10sequence of reactions that typically starts with a proton capture by a 12 C nucleus, as follows:!
12 C + 1 H → 13 N + γ

13 N → 13 C + e+ + ν

13 C + 1 H → 14 N + γ" 14

N + 1 H → 15 O + γ

15 O → 15 N + e+ + ν

15 N + 1 H → 12 C + 4 He

→ 16 O + γ

16 O + 1 H → 17 F + γ

17 F → 17 O + e+ + ν

17 O + 1 H → 14 N + 4 He

For example, if we denote by rpD the rate of 2 H + 1 H, one has rpD = rpp , etc. Note that describing the p-p reaction
Textbook by Prialnik
Textbookby Pols21‘slow’ and the H + H as ‘fast’ refers to the difference in cross-section factors #συ$ and not to the number of reactions pti1(6.4