Inside Living Stars, and Dead & Failed Stars

Inside Living Stars, andDead & Failed Stars

The instability stripHelioseismology

The standard solar modelElectron and Neutron Degeneracy Pressure

White Dwarfs and Neutron StarsBrown Dwarfs and Giant Planets

February 25, 2021

University of Rochester

Dead & Failed StarsI Pulsating stars and the instability stripI HelioseismologyI The standard solar modelI Degeneracy pressure of electrons and

neutronsI White dwarfsI Neutron starsI Brown dwarfs and giant planets

Reading: Kutner Ch. 10.4 & 11.2–11.3,Ryden Sec. 18.1–18.2

Right: HST image of the A1 star Sirius A (overexposed incenter) and its white-dwarf companion Sirius B (lower left).From NASA/HST.

February 25, 2021 (UR) Astronomy 142 | Spring 2021 2 / 36

https://www.nasa.gov/multimedia/imagegallery/image_feature_468.html

Patterns of pulsators: The instability strip (IS)

From Wikimedia commons.

Pulsating stars are not spread randomly in theluminosity-temperature space of the H-R diagram,nor are they all of a certain composition.I Mira variables are supergiants on the asymptotic

giant branch

I b Cep stars are all B stars lying just above themain sequence (MS).

I All the others lie along a long, narrow, nearlyvertical patch of the H-R diagram at effectivetemperatures ⇠ 104 K; this is the instability strip.

I d Scu stars are close to the main sequence butmost IS inhabitants are not there long; theydevelop rapidly after leaving the MS.


https://commons.wikimedia.org/wiki/File:HR-vartype.svg

Nonradial stellar oscillations

I Some pressure waves mayhave transverse components aswell as radial components.

I Such waves can be trappedbetween the surface and alayer of certain density in theinterior as they propagate.

I At the surface, the wavesreflect due to the densitydrop-off.

I In the interior, the wavesrefract, bending up due to theincrease of vs with depth.

Propagation of acoustic waves corresponding to modes with ` = 30 andn = 3 mHz (deeply penetrating rays) and ` = 100, n = 3 mHz(shallowly penetrating rays) (Christiansen 2003).


http://astro.phys.au.dk/~jcd/oscilnotes/

Nonradial stellar oscillations

I The deeper the waves penetrate,the fewer surface reflections.

I This can still make standing waves,and thus stable oscillations, byhaving an integer number ofwavelengths per circuit.

Modes with n = 3 mHz in order of decreasingpenetration depth: ` = 0, 2, 20, 25, and 75(Christiansen 2003).


http://astro.phys.au.dk/~jcd/oscilnotes/

Solar oscillations & helioseismology

The Sun oscillates in many different modes with periods of ⇠5 min. This was discoveredin 1962 (Leighton et al. 1962) and later explained as nonradial oscillations (Ulrich 1970).I It took a long time to notice these oscillations becuse they are so small. The velocity

amplitudes are typically 100 cm/s with displacements of tens of meters.I Many thousands of modes have now been identified.I The period of each mode gives the integral of

1vs

=r

r

gP

along a different path through the Sun’s interior.I Different modes penetrate more deeply into the interior, providing good coverage for

the outer 90% of the volume.


http://adsabs.harvard.edu/abs/1962ApJ...135..474L

http://adsabs.harvard.edu/abs/1970ApJ...162..993U

The solar oscillations

The Sun’s nonradial oscillations observed on a 12-second cadence. From SOHO/MDI (NASA)


http://soi.stanford.edu/results/greatest_hits.html

Precise knowledge of the solar interior

To summarize, we have:I A direct measure of pp fusion per unit time at the very center of the Sun using

measurements of solar neutrinos, corrected for neutrino oscillation.I Thousands of measurements of integrals of

1vs

=r

r

gP

from which the density, pressure, temperature, abundances, etc. can be determinedover most of the solar volume.

I Confidence that we know precisely and accurately the thermodynamic parameters ofall parts of the solar interior without being able to see the interior directly.

I The result is the Standard Solar Model (see, e.g., Bahcall et al. 2004).


http://adsabs.harvard.edu/abs/2005ApJ...621L..85B

The Standard Solar Model

P(r) and T(r), from Carroll and Ostlie, An Introduction to Modern AstrophysicsFebruary 25, 2021 (UR) Astronomy 142 | Spring 2021 9 / 36


r(r) and M(r), from Carroll and Ostlie, An Introduction to Modern AstrophysicsFebruary 25, 2021 (UR) Astronomy 142 | Spring 2021 10 / 36


Fusion power generation, from Carroll and Ostlie, An Introduction to Modern AstrophysicsFebruary 25, 2021 (UR) Astronomy 142 | Spring 2021 11 / 36


Radiation and convection zones, showing the dominant energy transport mechanism for each region, from Carroll andOstlie, An Introduction to Modern Astrophysics


How stars support themselves against gravity

Thermodynamics Gas and radiation pressure, for which the equations of state are

Pgas =rkTµ

= nkT Prad =4sT4

3c

support stars in which thermonuclear energy generation occurs.

Quantum Mechanics Degeneracy pressure sets in under extreme states of compressionand/or low temperatures. As we will see, this implies quite differentequations of state.I Degeneracy pressure is the means of support for objects with no internal

energy generation: dead stars like white dwarfs and neutron stars,unsuccessful stars (brown dwarfs), and the cores of giant planets(Jupiter, Saturn, etc.).


Degeneracy pressure

Degeneracy (or “exclusion”) pressure (Fowler 1926) is due to:

1. The Pauli Exclusion PrincipleI No two identical fermions (spin-1/2 particles) can be in the same quantum state

simultaneously.

2. The Heisenberg Uncertainty PrincipleI The uncertainties in the position and momentum of a particle are related by the

inequality

DxDpx � h̄2

I An electron confined to a small region Dx has a relatively large uncertainty in itsmomentum Dpx.


http://adsabs.harvard.edu/abs/1926MNRAS..87..114F

https://en.wikipedia.org/wiki/Pauli_exclusion_principle

https://en.wikipedia.org/wiki/Uncertainty_principle

Degeneracy pressureFrom these two principles we can make a hand-waving, non-rigorous derivation of theequation of state for degenerate matter. (See Introduction to Quantum Mechanics byGriffiths (Sec. 5.3.1) for a better derivation.)

Consider identical fermions of number density n. Each is inside its own little “box” ofspace with volume `3, such that

` = (1/n)1/3 length of boxDx = ` each particle is somewhere inside its box

If the particles have finite momentum, they hit the “walls” of their “boxes,” which wewill say have area A. Then they exert a pressure

P ⇡ FA

=1A

dpdt

=1A

nAdx2

| {z }# of pclshittingwall

1dt|{z}intervalof hit

px

|{z}typicalmomentumper pcl.

=12

nvxpx


Degeneracy pressureTypical momentum per particle:

px ⇡ Dpx not exactly, but within a factor ⇠ 1

⇡ hDx

by the Uncertainty Principle

= hn1/3 cube bounding each particle

Since the particles are moving in nonrelativistic motion, px = mvx and so

P =12

nvxpx =np2

x2m

=h2n5/3

2m

Done correctly using the Fermi-Dirac distribution for p, v, etc. gives the correct leadingfactor:

P = 0.0485h2n5/3

m


https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics

Degeneracy pressure in the labThe cause of the Pauli Exclusion Principle and its physical limits are not understood, butit is well-demonstrated experimentally.

I Left cloud: 6Li spin-1 bosons

I Right cloud: 7Li spin-3/2 fermions

I As the temperature drops, the bosonsbunch together. . .

I . . . but the cloud of fermions does notcondense below 300 nK.

I This is degeneracy pressure at work,driven by the Pauli Exclusion Principle

From Truscott & Hulet (2010)February 25, 2021 (UR) Astronomy 142 | Spring 2021 17 / 36

http://apod.nasa.gov/apod/ap100228.html

Electron degeneracy pressureConsider a gas of electrons, produced by ionization from atoms with nuclear charge Zeand baryon number A. The electron and nuclear densities are related by

ne = Zn+

and the mass density is

r = mene + Ampn+ (mp ⇡ mn)

⇡ Ampn+ (mp � me)

Therefore, ne = Zr/Amp and the equation of state for electron degeneracy pressure is

Pe = 0.0485h2

me

✓ZA

◆5/3 ✓ r

mp

◆5/3


Polytropic equation of state

Note that the equation of state is polytropic:

P µ rg

where g = (n + 1)/n is called the polytropic exponent and n is called the polytropicindex. This is one of the few functional forms for which the equations of stellar structureyield simple solutions. Example models:

Index n Notes

0.5 � 1 Neutron stars1.5 Fully convective cores, brown dwarfs, gaseous planets3 MS stars (Sun) and relativistic degenerate cores (WDs)

As n increases, the density distribution is more heavily weighted toward r = 0.


https://en.wikipedia.org/wiki/Polytrope

Electron degeneracy in stars

Suppose we use the new EOS (equation of state) instead of the ideal gas law to balancegravity and support a star. Our former scaling relationships derived from the ideal gaslaw and gravity were

Pc µGM2

R4 rc µMR3

Precise calculations for the polytropic EOS P µ r5/3 and gravity turn out to give

Pc = 0.77GM2

R4 rc = 1.43MR3

We can now use this to estimate the typical size of a star balanced by electron degeneracypressure.


Size of an electron degenerate star

Pc = Pe

0.77GM2

R4 = 0.0485h2

me

✓ZA

◆5/3 r5/3

m5/3p

= 0.0485h2

me

✓ZA

◆5/3 ✓1.43Mmp

◆5/3 1R5

) R = 0.114h2

Gmem5/3p

✓ZA

◆5/3M�1/3

A couple of points:I There is no temperature dependence here! This is much more simple than a normal

star.I R is much smaller than a normal star of equal mass.


White dwarfs

Numerical example

For M = 1M� and Z/A = 0.5, R = 9 ⇥ 108 cm ⇡ 1.4R�. This object, a white dwarf, hasthe mass of a star and the size of a planet!

I Remarkable feature of R–M relation: Rdecreases with increasing M (Stoner1930).

I Cause: larger mass M requires largersupporting Pe

I Larger Pe implies larger electronmomenta p confined to a smaller “box”Dx. The entire object is smaller!


http://adsabs.harvard.edu/abs/1930LEDPM...9..944S


Earth compared to the white dwarf star Sirius BFrom NASA+ESA.


http://www.spacetelescope.org/images/heic0516c/

Special Relativity & the white dwarf mass limit

I To support higher mass (smaller) white dwarfs, larger electron momenta (andspeeds) are required.

I Special Relativity: electron velocities cannot exceed c ⇡ 3 ⇥ 1010 cm/s.I Therefore, there is an upper bound (a maximum) to the mass that can be achieved by

a white dwarf (Anderson 1929, Stoner 1930, Chandrasekhar 1931).I Note: when v ⇠ c, p is not simply mv. In this extreme relativistic limit,

Pe = 0.123hcn4/3e

I Note that the relativistic and nonrelativistic expressions for Pe are equal atne = 1030 cm�3, about the density of the core of a 0.3M� white dwarf (Stoner 1930).


https://link.springer.com/article/10.1007/BF01340146


http://adsabs.harvard.edu/abs/1931MNRAS..91..456C


Stoner-Anderson-Chandrasekhar mass (“Chandrasekhar limit”)

For P µ r4/3 and gravity, the central pressure and density of a star are

Pc = 11GM2

R4 rc = 12.9MR3

Balance Pc with the relativistic electron degeneracy pressure Pe and the radius disappearsfrom the equation (you will show this in your homework). The result is a maximum massknown as the Stoner-Anderson-Chandrasekhar mass or the Chandrasekhar limit:

MSAC = 0.2✓

ZA

◆2

hcGm2

p

!3/2

mp = 1.44M�

for Z/A = 0.5.


Theory vs. observation for white dwarfsWhite dwarfs are hard to detect in binariesbecause they are so much fainter thanmain-sequence stars.I Blue squares show white dwarfs in binaries

with precisely known orbits (Provencal1998, Provencal 2002, Parsons 2010, Parsons2011, Parsons 2012a, Parsons 2012b, Pyrzas2012, Hermes 2012).

I The white dwarfs in visual binaries arelabeled with their names.

I The “best” white dwarfs are joined withothers detected in WD-MS systems by theSloan Digital Sky Survey (Rebassa 2012) andshown to be eclipsing (Parsons 2013).


http://adsabs.harvard.edu/abs/1998ApJ...494..759P



http://adsabs.harvard.edu/abs/2010MNRAS.402.2591P

http://adsabs.harvard.edu/abs/2011ApJ...735L..30P

http://adsabs.harvard.edu/abs/2011ApJ...735L..30P



http://adsabs.harvard.edu/abs/2012MNRAS.419..817P


http://adsabs.harvard.edu/abs/2012ApJ...757L..21H

http://adsabs.harvard.edu/abs/2012MNRAS.419..806R


Neutron starsWhat happens in a dead star with M > MSAC?I Such a star simply cannot be supported by electron degeneracy pressure. Add a little

too much mass and it will collapse gravitationally or explode.I During the collapse, the extra energy liberated from gravity, plus the high density,

can help drive some endothermic nuclear reactions, notably

DE + e� + p ! n + ne

I But neutrons are fermions too, and neutron degeneracy pressure can also balancegravity. A neutron star is formed (Tolman 1939, Oppenheimer 1939).

I The nonrelativistic R-M relation for a NS is

R = 0.0685h2

Gm8/3p

M�1/3

This is the mass of a star and the size of a city.February 25, 2021 (UR) Astronomy 142 | Spring 2021 27 / 36

http://adsabs.harvard.edu/abs/1939PhRv...55..364T

http://adsabs.harvard.edu/abs/1939PhRv...55..374O

Size of a 1M� neutron star


Pulsars

I Most of the neutron stars we observeappear to us as pulars.

I A pulsar is a rapidly rotating NS (orWD) that emits a beam ofelectromagnetic radiation.

I Neutron stars can also be observed asX-ray flaring members of low-massbinary systems (LMXBs).

I PS B1919+21 was the first pulsar to bediscovered in 1967 (Hewish et al. 1968)

I Right: the center of the Crab Nebula(M1), the remnant of SN 1054.

Optical/X-ray overlay from the Hubble and Chandratelescopes.


http://adsabs.harvard.edu/abs/1968Natur.217..709H

Pulsars

X-ray images of the Crab Nebula from the Chandra X-ray Observatory (CXO) taken 0.015 s apart.


http://chandra.harvard.edu/index.html

Neutron star masses

I The maximum mass of a neutron star is not easy to calculate as it requires GeneralRelativity and a strong-interaction equation of state.

I The maximum mass turns out to be about 2.2M� (Lattimer 2012).I The maximum cannot be > 3M� or the equation of state would imply a sound speed

vs > c.I A handful of neutron stars are observed in low-mass X-ray binary (LMXB) systems

and their masses and radii have been measured (see e.g. Steiner et al. 2010).I Neutron stars are so small that eclipses enabling the determination of R from timing

are extremely rare. Instead, we get R from the luminosity and temperaturedetermined from spectra taken in between flares (“quiescent LMXB”).


http://adsabs.harvard.edu/abs/2012ARNPS..62..485L

http://adsabs.harvard.edu/abs/2010ApJ...722...33S

Neutron star R–M relation

I The theoretical calculation of theNS R–M relation is beyond thescope of an undergraduate course.

I Currently the calculation ofLattimer & Prakash (2007) comesclose to matching the data onneutron star binaries (Steiner et al.2010).

I Note that circumference is plottedinstead of radius.


http://adsabs.harvard.edu/abs/2007PhR...442..109L



Degenerate star R–M relations

Gray line: causality cutoffA star smaller than 5.8pGM/c2

turns out to require an EOS with

∂P∂r

> c2

which in turn implies a speed ofsound vs > c, violating the order ofcause and effect.


Degenerate star R–M relations

I Gray line: vs > cI Dark gray line: P > •I Black line: event horizon

(would be a black hole)

In all these systems, the very largemasses and small distances requirea General Relativistic treatment.


Far below MSAC: Brown dwarfs and giant planets

When stars form, they contract until they are hot enough in the center (about 3 ⇥ 106 K)to ignite the pp chain fusion reactions. Recall that

Tc =Pcµc

rck⇡ µc

k

✓GM2

R4

◆✓1

150R3

M

◆

= 15.7 ⇥ 106✓

MM�

◆✓R�R

◆K

for solar-type stars, if gravity is supported by gas pressure.I For small masses this involves gas pressures that become smaller than the electron

degeneracy pressure, so degeneracy pressure can stop the contraction and preventthe object from reaching fusion temperatures. This imposes a lower mass limit onwhat can become a star.

I The H-burning limit turns out to be 0.08M�.


Far below MSAC: Brown dwarfs and giant planets

Depending upon how they are formed and what their mass is, such objects are calledbrown dwarfs or giant planets.I Because they cannot replace the energy that leaks away in the form of radiation, they

simply remain at the size determined by degeneracy pressure and cool off forever.I Thus, if they are very old, they are very faint. This prevented their detection until

1995.I Today, thousands are known from deep near-IR surveys and from Spitzer Space

Telescope observations.I Once it was thought that these objects could be numerous enough to comprise a

significant (and invisible) component of the mass in the Galaxy.I We will come back to this idea when we discuss Dark Matter.


Inside Living Stars, and Dead & Failed Stars

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