The Red Giant Branch. L shell drives expansion L shell driven by M core - as | |, | T| increase outside contracting core shell narrows, also L core

The Red Giant Branch


• Lshell drives expansion

• Lshell driven by Mcore - as ||, |T| increase outside contracting core shell narrows, also Lcore from contraction increases Tshell

• Lshell large, rshell small so convection necessary

• 1st dredge-up - envelope convection zone reaches material processed by H burning











The Red Giant Branch-Low Mass Stars

• e- degeneracy

consider e- in a boltzmann distribution in phase space

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f ( p)dpdV = ne

4πp2

(2πmekT)3 2e

−p 2

2me kT dpdV

f ( p) =#e− in [ p, p + dp]

ne = f ( p)dp ;0

∞

∫ pmax = (2mekT)1 2


max occupancy of phase space from Pauli exclusion

volume of phase space cell dxdydzdpxdpydpz=h3

so in [p,p+dp] 4dpdV/h3 cells each with max occupancy of 2e- (spin ,)

at low T or high ne distributions diverge from boltzmann due to occupancy of available states

if all e- have lowest possible energy

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f ( p) =8πp2

h3p ≤ p fermi

f ( p) = 0 p > p fermi

nedV = dV8πp2

h3dp

0

p f∫ =8π

3h3p f

3dV

p f =3h3ne

8π

⎛

⎝ ⎜

⎞

⎠ ⎟

1

3

E f =p f

2

2m3


all available states populated up to pf so for high ne vfc

€

p = γmev

E tot = γmec2 = mec

2 1+p2

mec2

⎛

⎝ ⎜

⎞

⎠ ⎟

1 2

1

c

∂E tot

∂p=

p mec

1+ p2 me2c 2

[ ]1 2 = β

E = E tot − mec2

Pressure = p flux through unit surface s

flux through d w/ [p,p+dp]

d

d€

ˆ s

€

ˆ n

€

f ( p)dpdΩ /4π at dσ

flux = f ( p)dpdΩv(p)cosθdσ /4πr p ⋅ ˆ n = pcosθ


€

Pe ( ˆ n ) = f (p)v(p)pcos2 θdpdΩs /4π0

∞

∫ =8π

3h3p3v( p)dp

0

p f∫2π

∫

Pe =8πc

3h3p3 p /mec

1+ p /me2c 2

[ ]1/ 20

p f∫

=8πc 5me

4

3h3

ξ 4dξ

(1+ ξ 2)1/ 20

x

∫ x =p f

mecξ =

p

mec

Pe =πme

4c 5

3h3x(2x 2 − 3)(x 2 +1)1/ 2 + 3ln[x + (1+ x 2)1/ 2]

ne =8πme

3c 3

3h3x 3

Ue = f ( p)E( p)dp =πme

4c 5

3h38x 3[(x 2 +1)1/ 2 −1] − f (x)

0

p f∫


Relativistic vs. non-relativistic

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x =p f

mec= γβ fermi

for x<<1 - non-relativistic

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Pe =2πme

4c 5

3h3x 4 =

3

π

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 3hc

8ne

4 / 3 =1.2435 ×1015 ρ

μ e

⎛

⎝ ⎜

⎞

⎠ ⎟

4 / 3

=1

3Ue

€

Pe =8πme

4c 5

15h3x 5 =

1

20

3

π

⎛

⎝ ⎜

⎞

⎠ ⎟2 / 3

h2

me

ne5 / 3 =1.0036 ×1013 ρ

μ e

⎛

⎝ ⎜

⎞

⎠ ⎟

5 / 3

=2

3Ue

for x>>1 - relativistic


Meanwhile, back in the star…

• Stars < ~2.25 M have lower Tcore and lower entropy (higher for a given T)

• Low T combined with high ne mean core becomes degenerate before reaching He burning T

• degenerate cores reach Tignition (~2e8 K) at 0.46 M

• L Mcore so L is ~ the same for all stars which undergo degenerate He ignition - max L of RGB for old clusters

• Tip of the RGB method for getting distance


When degenerate stars reach T~2x108K• Core is roughly isothermal, so a large volume is close to ignition• P is not proportional to T since pressure is from degeneracy T from burning does not result in explosive burning

The Red Giant Branch-Low Mass StarsHe flash• Explosive burning of He to 12C - not energetic enough to disrupt star,

but may result in a puff of mass loss• Energy release heats core until degeneracy is lifted - normal HSE

resumes• Hydrostatic He burning: triple process

(2,)12C (,)8Be stable by only 92keV

lifetime of excited state <<mean collision time unless there is a resonanceHoyle predicts resonant energy level in 8Be(,)12C, confirmed by nuclear physics experiments

note 2 - 3 body reaction so very density sensitive -reason #1 for big bang nucleosynthesis cutoff

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≈h/ΔE = 2.6 ×10−16 s

€

ε3α ≈ ε1ρ2Xα

3 T8

2

⎛

⎝ ⎜

⎞

⎠ ⎟

18.5

ε1 = 23.1erg g−1 s−1

The Red Giant Branch-Low Mass StarsHydrostatic He burning part II

• 12C(,)16O• rate uncertain - too high and all He O; too low and C/O too

high• at low Y12Cmostly (2,)12C• as Yhe drops 12C(,)16O dominates due to Y3

He dependence• So Y12C sensitive to ingestion of He at late times• also sensitive to entropy - 3 rate 2 so lower at high S

more massive stars have higher 16O/12C• 16O(,)20Ne slow at these temperatures• 14N(,)18O depletes N very rapidly

– 18O(,)22Ne22Ne(,)26Mg22Ne(,n)25Mg - neutron source

Post-RGB Evolution - Low Mass

Once hydrostatic He burning has begun in the core

• Core expands, envelope contracts - Lsurf R

• Blue loops

1. RGB

2. He flash

3. Max extent of blue loop - Xhe ~ 0.1


Extent of blue loop depends on

1. metallicity - low z


Extent of blue loop depends on

1. metallicity - low z large blueward excursion

2. core size (initial M) - higher mass large blueward excursion

3. mixing and EOS influence max Teff

Blue horizontal branch


Distance between subgiant branch and horizontal branch used as proxy for cluster age - depends only on composition & age - insensitive to reddening

Width of subgiant branch also used - for clusters w/ poorly populated HB

Cepheids

• Stars of ~4 M move far enough to the blue on the horizontal branch to enter a region of instability

• This strip extends to much lower luminosities and crosses the main sequence producing Scuti stars

Cepheids

The mechanism• Opacity will be large at temperatures close to the

ionization temperature of H and He.• Ionized material has high opacity, opacity drops

precipitously upon recombination

Cepheids

The mechanism• Opacity will be large at temperatures close to the

ionization temperature of H and He.• Ionized material has high opacity, opacity drops

precipitously upon recombination• Radiation pressure on a high region causes it to

expand and cool• Sufficient expansion cools material enough for

recombination sharp • Pressure supports goes away and region contracts

and heats, reionizing material - Carnot engine• Pulsations occur only if not damped by too much

mass above proper T, also must have enough mass to provide restoring force - hence instability strip

Documents

The Red Giant Branch. L shell drives expansion L shell driven by M core - as | |, | T| increase outside contracting core shell narrows, also L core