Virial Theorem dand

Stellar StructureStellar Structure

Star produces energy via nuclear fusion

mass： 2×1030 kg

Sun：our nearest star

radius： 7×105 km(109×Earth radius)( )

N l f i f H d 6 1×1011 k dNuclear fusion of Hydrogen: 6.1×1011 kg per second 3.85×1026J / seccf. a nuclear power plant produce 109 J / sec

Stellar Structure静⽔圧平衡 （せいすいあつへいこう、hydrostatic equilibrium）

重⼒による収縮 と 圧⼒勾配による膨張が釣り合った状態Gravity balances with Pressure Gradient

What is pressure? L

# A particle collides with a wall perpendicular to x-axis.It gives momentum （運動量） to the wall

N particles

2 ( 2 )p(px,py,pz)

# A particle collides with the wall per unit time v(vx,vy,vz)

2px (=2mvx)

x# A particle gives momentum to the wall per unit time

vx/(2L) times

# Momentum given to a wall per unit time per unit area

2px ×vx/(2L) = px vx /L

g p p＝Force given to a wall per unit area＝pressure Total kinetic energy

of gas particles in a box

VEmvnvpn

LLvpNP KExx

32

21

32

311 2

2

“p” is momentum 運動量“P” is pressure

number density

Pressure Gradient

Diff ( di t) f t d + Positive = outward directionDifference (gradient) of pressure at r and r+r

ArdrdPAr

drdPrPrPArrPrP

)()()()(

Positive outward direction

drdr

rRr

rR

M=(r)rA

PA (P+P)A

M (r)rAArea A

Density and temperature is higher at inner radii 0dPDensity and temperature is higher at inner radii 0dr

Force by pressure gradient … positive = outward direction

Particle A pulls particle B by gravitational force

Gravity

mA mBr

Particle A pulls particle B by gravitational force

mGm BA2r

Gravitational energy of particle BmGm BAgy p

r

r

2

density

rRGas mass inside radius r

It gives gravity to■

rdrrrm 2

0

4)()(

M=(r)rAIt gives gravity to ■

])([)()( ArrrGmMrGm ])([22 Arr

rM

r

Balance between gravity and pressure gradient

0))(()( ArdPArrrGm

Gravity balances with Pressure Gradient

0))((2 Ardr

Arrr

dPrrGm )()( drd

rGm

2

)()(

rrR

r

PA (P+P)A

M=(r)rAArea A

Gravity: inwardsPressure gradient: outwards

Virial Theorem and Hydrostatic Equilibrium

2

)()(r

rrGmdrdP

Hydrostatic Equilibrium：

Multiply by 4 r3 and integrate from r = 0 to r=R

RR

drrrGmddP 23 4)()(4

=dm rdm dr

rdr

drr

00

3 )()(4

M

dm

Right term ＝ M

gravEdmr

rGm

0

)(

Left term＝ drrdrrrP

drrrPrrPR

R

RR

20

2

23 44)(

304)(3]4)([

Left term＝ drrdrr

drrrPrrP R

0

0

200 4

4304)(3]4)([

= - 3<P>VE1 0

VE

P grav

31

weighted meanMean pressure weighted by volume

Virial Theorem and Hydrostatic Equilibrium

VE

P grav

31

Mean pressure in a starV3

VEP KE

32

SubstituteV

EV

E KEgrav

32

31

V3 VV 33

Virial Theorem

02 gravKE EE1

gravKEgravKETOT EEEEE21

Application of Virial Theorem

02 gravKE EE

gravKEgravKETOT EEEEE21

・ETOT < 0 …bound

2

・When energy is lost (ETOT decreased）, temperature rises （EKE increased）

Photon is emitted from stellar

example：Kelvin-Helmholtz Contraction

Photon is emitted from stellar surface Energy is lost

ETOT & E decrease

Gas contractsto get hotter

ETOT & Egrav decrease EKE increase

Contraction stopswhen the temperature and density get highenough for nuclear fusion

Starルツシ プルング ラ セル図

brヘルツシュプルング－ラッセル図Surface temperature vs brightness 重い

right

巨星惑星状星雲

Most stars are plotted along a lineMain sequence...Main sequencenuclear fusion of Hydrogen to Helium

Sun

Massive MS stars are rareformation rate is low

軽いspectrum type mass life time [yr]

B 20 107

... formation rate is lowshort lifetime

faint

high← temperature → low

B 20 107

A 3 5×108

G 1 1010

spectrum typeK 0.6 1011

Main Sequence star is stable

02 gravKE EE1

gravKEgravKETOT EEEEE21

emission＞ nuclear fusion E decreases ETOT decreases Egrav decreases（contraction） EKE incteases（temperature rise） h l f i enhances nuclear fusion