IUCAA-NCRA Graduate School 2016dipankar/gradschool2016/gradschool2016_stars.pdf · 1011 stars 1987A&A...184..104R Rana 1987. ... before the C-ignition temperature is reached and becomes

IUCAA-NCRA Graduate School 2016 Introduction to Astronomy and Astrophysics part II (Stars)

Tue, Thu 10 AM - 11 AM; Wed 11.30PM- 12.30 PM 14 lectures

Lecture dates:October 19,20,25,26,27November 2,3,8,10,15,17,22,23,24

Evaluation:Assignment 1: Handout Oct 25, due Nov 2 (15%)Assignment 2: Handout Nov 17, due Nov 24 (15%)Mid-term Examination: November 9 1130-1300 (30%) [open notes, text]Final Examination: December 1 0930-1100 (40%) [closed book]

Primary Texts:Dina Prialnik: An Introduction to the theory of stellar structure and evolutionR. Kippenhahn & A. Weigert: Stellar Structure and Evolution

http://www.iucaa.in/~dipankar/gradschool2016/

http://www.iucaa.in/~dipankar/gradschool2016/

Basic observables of a star in optical band

• Apparent Magnitude mV• Colour Index B-V ≡ mB - mV • Spectra • Parallax distance d

‣ Hipparcos (1989-93) : up to 500 pc‣ Gaia (2013-) : up to 10 kpc

• Mass - from binary systems

Absolute Magnitude MV = mV - 5 log10 (d / 10 pc) [Measure of Luminosity]

Colour Index: Measure of Temperature

HipparcosH-R diagram


Main Sequence


Main Sequence

Effective Temperature

Solar Spectrum

A typical stellar spectrum is nearly a blackbody

Total Luminosity

Color Temperature= Temp. of best-fit Planck Function

Te↵ is defined as the Effective Temperature

L = 4⇡�T 4e↵R

2

Colour -Temperaturerelation


Main Sequence

O B A F G K M

3000

0 - 5

0000

K

1000

0 - 3

0000

K

7500

- 10

000

K

6000

- 75

00 K

5000

- 60

00 K

3500

- 50

00 K

2000

- 35

00 K

Stellar Spectral Classification

http://azastro.pbworks.com

http://azastro.pbworks.com

1987A&A...184..104R

Mass-Luminosity relation on Main Sequence

Rana 1987

A typical galaxylike the Milky Waycontain about1011 stars

1987A&A...184..104R

Rana 1987

Mass functionat birth

Salpeter 1955:

dN/d(log M)∝ M-1.35

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Thursday, March 18, 2010

Adelberger et al Rev Mod Phy 83, 195 (2011)

Theoretical H-R diagram

O B A F G K MStellar evolutionary tracks by Schaller et al 1992

Stellar spectral classification

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Thursday, March 18, 2010

Adelberger et al Rev Mod Phy 83, 195 (2011)

PH217: Aug-Dec 2003 1

Equations of Stellar Evolution

The complete set of differential equations of stellar evolution are as follows:

∂r∂m =

14πr2ρ

, (1)

∂P∂m = − Gm

4πr4 −1

4πr2∂2r∂t2 , (2)

∂T∂m

= − GmT4πr4P

∇ , (3)

∂L∂m

= ϵn − ϵν − CP∂T∂t+δρ∂P∂t, (4)

∂Xi

∂t =mi

ρ

⎛⎜⎜⎜⎜⎜⎜⎝∑

j

r ji −∑

k

rik

⎞⎟⎟⎟⎟⎟⎟⎠ , i = 1, . . . , I. (5)

where explicit time dependence is retained for all quantities. In eq. (4) ϵn isthe nuclear energy generation rate per unit mass at lagrangian coordinatem, ϵν the energy loss rate due to neutrino emission, δ ≡ −(∂ lnρ/∂ ln T)P. Ineq. (5) Xi is the mass fraction of nuclear species i, while the total number ofrelevant nuclear species is I. ri j is the rate of transmutation (reaction rate)of species i into species j. In eq. (4) the last two terms can be combined andwritten in terms of entropy s as −T∂s/∂t.

The physical properties of the stellar material ρ, CP, δ, ∇ad, opacity κ, ϵn, ϵνand ri j can all be expressed as functions of pressure P, temperature T andcomposition Xi (i = 1, . . . , I). Thus the equations eq. (1) – eq. (5) form a setof 4 + I equations in 4 + I variables r,P,T,L,X1, . . . ,XI, and can thereforebe solved, subject to suitable boundary conditions, to yield the structureof the star at any stage of evolution. One seeks a solution in the range 0 ≤m ≤M, where M is the total mass of the star. Numerical computations arerequired to solve the coupled set of equations and obtaining the structureand evolution of the star.

The role of different time scales are clear from the above equations. Thetime derivative in eq. (2) defines the hydrodynamic time scale τhydr. Timederivatives in eq. (4) correspond to the Kelvin-Helmholtz time scale τKH.

Equations of stellar evolution

PH217: Aug-Dec 2003 1

Equations of Stellar Evolution

The complete set of differential equations of stellar evolution are as follows:

∂r∂m =

14πr2ρ

, (1)

∂P∂m = − Gm

4πr4 −1

4πr2∂2r∂t2 , (2)

∂T∂m

= − GmT4πr4P

∇ , (3)

∂L∂m

= ϵn − ϵν − CP∂T∂t+δρ∂P∂t, (4)

∂Xi

∂t =mi

ρ

⎛⎜⎜⎜⎜⎜⎜⎝∑

j

r ji −∑

k

rik

⎞⎟⎟⎟⎟⎟⎟⎠ , i = 1, . . . , I. (5)

where explicit time dependence is retained for all quantities. In eq. (4) ϵn isthe nuclear energy generation rate per unit mass at lagrangian coordinatem, ϵν the energy loss rate due to neutrino emission, δ ≡ −(∂ lnρ/∂ ln T)P. Ineq. (5) Xi is the mass fraction of nuclear species i, while the total number ofrelevant nuclear species is I. ri j is the rate of transmutation (reaction rate)of species i into species j. In eq. (4) the last two terms can be combined andwritten in terms of entropy s as −T∂s/∂t.

The physical properties of the stellar material ρ, CP, δ, ∇ad, opacity κ, ϵn, ϵνand ri j can all be expressed as functions of pressure P, temperature T andcomposition Xi (i = 1, . . . , I). Thus the equations eq. (1) – eq. (5) form a setof 4 + I equations in 4 + I variables r,P,T,L,X1, . . . ,XI, and can thereforebe solved, subject to suitable boundary conditions, to yield the structureof the star at any stage of evolution. One seeks a solution in the range 0 ≤m ≤M, where M is the total mass of the star. Numerical computations arerequired to solve the coupled set of equations and obtaining the structureand evolution of the star.

The role of different time scales are clear from the above equations. Thetime derivative in eq. (2) defines the hydrodynamic time scale τhydr. Timederivatives in eq. (4) correspond to the Kelvin-Helmholtz time scale τKH.

0.01

0.1

1

10

100

1000

10000

3.6 3.8 4 4.2 4.4

log

L/Ls

un

log Teff (K)

Zero Age Main Sequence

0.50.6

0.70.8

0.91.0

3.0

5.0

7.0

9.0

M / Msun

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1

log

rho

(g/c

m^3

)

(m/M)

ZAMS models

1 Msun9 Msun

3.5

4

4.5

5

5.5

6

6.5

7

7.5

0 0.2 0.4 0.6 0.8 1

log

T (K

)

(m/M)

ZAMS models

1 Msun9 Msun

-6

-5

-4

-3

-2

-1

0

0 0.2 0.4 0.6 0.8 1

log(

1-be

ta)

(m/M)

ZAMS models

1 Msun9 Msun

0.5 Msun

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

dlnT

/dln

P

(m/M)

ZAMS models

1 Msun9 Msun

0.5 Msun

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

1000 10000 100000 1e+06 1e+07 1e+08 1e+09 1e+10 1e+11

log

(R/R

sun)

, log

(L/L

sun)

Time (years)

Evolution of a 1 solar mass star

LuminosityRadius

ZAMS now

0

0.2

0.4

0.6

0.8

1

-4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

XH, X

He

log (m/M)

Solar Model

HHe

ZAMS

ZAMS

now

now

-2

-1

0

1

2

3

4

5

3.4 3.6 3.8 4 4.2 4.4 4.6

log

Lum

inos

ity (S

olar

Uni

ts)

log Teff (K)

Evolution of a 1 Solar Mass star

ZAMSSun

pre-MS

post-MS

Stellar Evolution

STARSA public domain code for computation of structure and evolution of single and binary stars

P. P. Eggleton et al, University of Cambridge

Download code, data and documentation from http://www.ast.cam.ac.uk/~stars/index.htm

Use a UNIX/Linux workstation, compile using gfortran

http://www.ast.cam.ac.uk/~stars/index.htm

0.5

0.7

1.0

3.0

5.0

7.0

9.0

M / Msun

-2

-1

0

1

2

3

4

5

3.4 3.6 3.8 4 4.2 4.4 4.6

log

Lum

inos

ity (S

olar

uni

t)

log Teff (K)

Stellar evolution: Solar Metallicity

-2

-1

0

1

2

3

4

5

3.4 3.6 3.8 4 4.2 4.4 4.6

log

Lum

inos

ity (S

olar

uni

t)

log Teff (K)

Stellar evolution: Solar Metallicity

0.5

0.7

1.0

3.0

5.0

7.0

9.0

M / Msun

Haya

shi tr

ack

7

7.5

8

8.5

9

0 1 2 3 4 5 6 7 8

log

cent

ral t

empe

ratu

re (K

)

log central density (g/cc)

7 Msun

1 Msun

He ignition

Degeneracy

T/TF ~

0.1

Figure 8.4. Detailed evolution tracks in the log ρc - log Tc plane for masses between 1 and 15M⊙. Theinitial slope of each track (labelled pre-main sequence contraction) is equal to 1

3 as expected from our simpleanalysis. When the H-ignition line is reached wiggles appear in the tracks, because the contraction is then nolonger strictly homologous. A stronger deviation from homologous contraction occurs at the end of H-burning,because only the core contracts while the outer layers expand. Accordingly, the tracks shift to higher densityappropriate for their smaller (core) mass. These deviations from homology occur at each nuclear burningstage. Consistent with our expectations, the most massive star (15M⊙) reaches C-ignition and keeps evolvingto higher T and ρ. The core of the 7M⊙ star crosses the electron degeneracy border (indicated by ϵF/kT = 10)before the C-ignition temperature is reached and becomes a C-O white dwarf. The lowest-mass tracks (1 and2M⊙) cross the degeneracy border before He-ignition because their cores are less massive than 0.3M⊙. Basedon our simple analysis we would expect them to cool and become He white dwarfs; however, their degenerateHe cores keep getting more massive and hotter due to H-shell burning. They finally do ignite helium in anunstable manner, the so-called He flash.

Suggestions for further reading

The schematic picture of stellar evolution presented above is very nicely explained in Chapter 7 ofPrialnik, which was one of the sources of inspiration for this chapter. The contents are only brieflycovered by Maeder in Sec. 3.4, and are somewhat scattered throughout Kippenhahn & Weigert, seesections 28.1, 33.1, 33.4 and 34.1.

120

I. Ib

en 1

985

QJRA

S 26,

1

12C+12C → 23Na+p ; 20Ne+α20Ne+20Ne+γ → 16O+24Mg+γ16O+16O → 28Si+α28Si+28Si → 56Fe+2e+

Core-collapse Supernovae:Etot ~ 1053 erg; Ekin ~ 1051 erg; Erad ~ 1049 ergr-process nucleosynthesisMassive, fast spinning stars ⇒ jets ⇒ GRB

C

O

Si

Ne

8.5

8.6

8.7

8.8

8.9

9

9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9

10

3 4 5 6 7 8 9 10

log(T c

)

log( c)

Fe disintegration

Pair instability

M 20He 8M 80He 36

ρR.

Wal

dman

200

8 Ap

J 685

, 110

3

1985QJRAS..26....1I

I. Ib

en 1

985

QJRA

S 26,

1

1985QJRAS..26....1I

I. Ib

en 1

985

QJRA

S 26,

1

1985QJRAS..26....1I

I. Ib

en &

A.V

. Tut

ukov

198

5 Ap

JS 58

, 661

Inverse β decay: neutronization

np

e

npe

Low density

High density

At high density inverse β decay depletes electrons and softens electron degenerate Equation of State

⌫

⌫

µn = µp + µe

Beta equilibrium:µ =

qp2

F +m2c4;

Equilibrium nuclear composition may bederived by combining with nuclear mass formula including shell effects

2

2.2

2.4

2.6

2.8

3

3.2

3.4

1e+06 1e+07 1e+08 1e+09 1e+10 1e+11 1e+12

A/Z

log density (g/cc)

Equilibrium Nuclear Composition below Neutron Drip

ρ (g/cc)

A / Z

Equilibrium nuclear composition

Pycnonuclear Reactions : XAZ + XA

Z = Y2A2Z

and electron capture cause transmutationsto produce equilibrium composition of ColdCatalysed Nuclear Matter ⇒

Baym, Pethick & Sutherland 1971

GR-limited neutron degenerate structure

6

8

10

12

14

16

18

20

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

Rad

ius

(km

)

Mass (solar units)

Mass-radius relation for neutron degeneracy alone

Oppenheimer-VolkoffLimit (1939): 0.71 Msun

dPdr=

G(M + 4�r3P/c2)(⇥ + P/c2)r2(1 � 2GM/rc2)

(TOV Equation)

Importance of nuclear forces

• In realistic neutron stars nucleons are squeezed within inter-particle distance of ~1 fermi

• Strong nuclear forces become important contributor to EoS

• Nature of this force still uncertain, hence uncertainty in EoS

V(r)

r0

1 fermi

Repulsive strong forces wouldraise the upper mass limit

Demorest et al 2010

Neutron Star Mass-Radius Relation

Nuclear DensityNuclear Density

Nuclear Density

Neutron Drip

Inne

r C

rust

Out

er C

rust

1.4 Msun APR98 dV UIX

superfluid n

superconducting p

Fast spin (P ~ ms to s)

Strong Magnetic Field

Highly active magnetosphere

Ultrarelativistic charged particles

Strongly beamed broadband radiation: Pulsars

M ~1.5 M⦿ , R ~10kmstrong surface gravityX-rays from accretion:X-ray Binaries

Physical processes during core collapse

Density > 1010 g/cm3; Temperature ~ 1010 KPlasma neutrinos generated in addition to neutronization products

T. J

anka

, MPA

Core collapse and bounce

Wik

iped

ia

SN II : H in spectra SN I : no H - Ic: No Si, No He Ib: No Si Ia: Si absorption at 615 nm

Core collapse

WD explosion

Supernovae & Gamma Ray Bursts

SN Spectralclassification

SN Lightcurves

origin

Talk

Orig

ins.o

rg

Core collapse SN: Etot ~ 1053 erg Ekin ~ 1051 erg

SN Ia: Etot ≈ Ekin ~ 1051 erg

Long GRBs associated with SN I b,c: BH formation, jet ejection

(binary)

Short GRBs from NS merger (?) NS collapse to BH, jet ejection, Kilonova, Gravitational Waves

http://TalkOrigins.org

2040

6080

100

20 40 60 80 100

−5

−4

−3

−2

−1

Roche potential

PH217: Aug-Dec 2003 1

Binary Stars

A large fraction of stars are born in binary systems. In close binaries withinitial orbital periods less than a few years, the tidal forces due to thecompanion can significantly influence the course of evolution of a star.

Let us consider two stars of mass M1 and M2 in a binary system with acircular orbit of separation a. In a frame corotating with the binary theforce at any point P can be derived from an e↵ective potential of the form

V = �GM1

r1� GM2

r2�⌦2

r

23

2(1)

where r1 and r2 are the distances of P from M1 and M2 respectively, and r3is its distance from the rotation axis, which passes through the centre ofmass. ⌦ =

pGM/a3 is the orbital frequency of the binary, the orbital period

being Porb = 2⇡/⌦. The distances r1, r2 and r3 can be written in terms ofthe coordinate ~r of the point P to obtain V as a function of ~r. This is calledthe “Roche potential”. If the point P is located in the orbital plane then r3is simply the distance from the centre of mass (see fig. 1).

Equipotentials of the Roche potential function in the orbital plane revealan interesting structure: there are three saddle points of the potential andtwo maxima, as shown in figure 2. These are called the Lagrangian pointsand are designated L1 to L5, as shown in the figure. The lobes through theL1 point that encircle the two stars are called the “critical potential lobes”or “Roche lobes”. In the course of evolution if one of the stars expandsand fills its Roche Lobe then matter would flow from it to the companion.Usually one expresses the size of the Roche lobe in terms of the criticalradius R

L

, defined as the radius of the sphere which has the same volumeas the Roche Lobe. An expression (given by Eggleton) for R

L

(around M1)accurate to within one percent for all mass ratios is as follows:

R

L

a

=0.49

0.6 + q

2/3 ln(1 + q

�1/3)

where q =M2/M1.

If both components of the binary system are smaller than their respective

PH217: Aug-Dec 2003 1

Binary Stars



V = �GM1

r1� GM2

r2�⌦2

r

23

2(1)





L


L


R

L

a

=0.49

0.6 + q

2/3 ln(1 + q

�1/3)

where q =M2/M1.


PH217: Aug-Dec 2003 1

Binary Stars



V = �GM1

r1� GM2

r2�⌦2

r

23

2(1)





L


L


R

L

a

=0.49

0.6 + q

2/3 ln(1 + q

�1/3)

where q =M2/M1.


PH217: Aug-Dec 2003 1

Binary Stars



V = �GM1

r1� GM2

r2�⌦2

r

23

2(1)





L


L


R

L

a

=0.49

0.6 + q

2/3 ln(1 + q

�1/3)

where q =M2/M1.


PH217: Aug-Dec 2003 2

M M1 2

r rr1 23

P

C.M.

Figure 1: The three distances r1, r2 and r3, in the orbital plane of a binarysystem of masses M1 and M2, which enter the expression for the RochePotential in a reference frame co-rotating with the binary. C.M. indicatesthe centre of mass of the binary.

Roche Lobes, then the binary is called “detached”. If one component fillsits Roche Lobe then the binary is called “semi-detached”, and when bothcomponents grow to fill their corresponding Roche Lobes together thebinary is said to be a “contact binary”. In this last situation an envelopemay form covering both the components. This is known as the “commonenvelope” phase of binary evolution. Usually a large amount of matteris lost from the binary through the L2 and L3 points during a commonenvelope phase.

Transfer of mass from one star to another would change the orbital param-eters of the binary system. The orbital angular momentum of the binarycan be written as

J =M1M2

rGa

M

(2)PH217: Aug-Dec 2003 4

where M = M1 +M2 is the total mass of the system. From this, one maywrite

a

a

= 2J

J

� 2M1

M1� 2

M2

M2+

M

M

(3)

If the evolution is “conservative”, namely that no matter or angular mo-mentum leaves the binary system, then both M and J vanish. Eq. 3 thenyields

a / 1(M1M2)2

ora

a

= �2M1

M1

✓1 � M1

M2

◆

where we have used M2 = �M1 due to mass conservation. If M1 is themass losing star, M1 is negative. We therefore see that if the donor is moremassive than the accretor, then the orbit will shrink, and if the donor isless massive than the accretor then the orbit will widen upon conservativemass transfer.

Since the size of the Roche lobe around the components are proportionalto the orbital separation a, shrinkage of the orbit also implies shrinking ofthe Roche Lobe. This can make the Roche-Lobe-Overflow (RLOF) masstransfer highly unstable when matter goes from the more massive to theless massive star. As seen in fig. 3, starting from first contact at the right-most point in the diagram, the size of the orbit, and hence the Roche lobebegins to decrease rapidly while the equilibrium radius of the star doesnot decrease so quickly. As a result the RLOF mass transfer rate becomesvery large and would finally stop only when the mass ratio is much pastreversal, i.e. the donor star has become much less massive than the accretorstar. At this point normally just the Helium core of the donor star is left,the rest of the matter having been transferred onto the accretor. This is theresolution of the classical Algol paradox, involving the binary star Algol inwhich the less massive component was found to be more evolved, contraryto the expectation from single star evolution.

When the mass donating star is less massive than the accretor, as canhappen when the accretor is a compact star: white dwarf, neutron star orblack hole, the Roche Lobe would expand upon mass transfer and mass

PH217: Aug-Dec 2003 4


a

a

= 2J

J

� 2M1

M1� 2

M2

M2+

M

M

(3)


a / 1(M1M2)2

ora

a

= �2M1

M1

✓1 � M1

M2

◆




PH217: Aug-Dec 2003 4


a

a

= 2J

J

� 2M1

M1� 2

M2

M2+

M

M

(3)


a / 1(M1M2)2

ora

a

= �2M1

M1

✓1 � M1

M2

◆




PH217: Aug-Dec 2003 4


a

a

= 2J

J

� 2M1

M1� 2

M2

M2+

M

M

(3)


a / 1(M1M2)2

ora

a

= �2M1

M1

✓1 � M1

M2

◆




PH217: Aug-Dec 2003 4


a

a

= 2J

J

� 2M1

M1� 2

M2

M2+

M

M

(3)


a / 1(M1M2)2

ora

a

= �2M1

M1

✓1 � M1

M2

◆




PH217: Aug-Dec 2003 4


a

a

= 2J

J

� 2M1

M1� 2

M2

M2+

M

M

(3)


a / 1(M1M2)2

ora

a

= �2M1

M1

✓1 � M1

M2

◆




::

PH217: Aug-Dec 2003 5

M M M M

R,RL

01 2c =

M1

Figure 3: Evolution of the Roche Lobe (red) and the radius (blue) of themass-donating star of mass M1 during conservative mass transfer in abinary. The accretor star (of mass M2) is less massive than the donor star atthe start of the mass transfer (rightmost point in the figure). Mass transfercauses M1 to decrease and M2 to increase, and the orbital separation andhence the Roche Lobe size passes through a minimum when the mass ratiobecomes unity. The evolution of the stellar radius ensures continuous,heavy mass transfer until only the Helium core (of mass M

c

) of the donoris left.

Orbital evolution due to Mass TransferPH217: Aug-Dec 2003 4


a

a

= 2J

J

� 2M1

M1� 2

M2

M2+

M

M

(3)


a / 1(M1M2)2

ora

a

= �2M1

M1

✓1 � M1

M2

◆




< 0 (donor)

D. Bhattacharya and E.P.J. van den Heuvel, Binary and millisecond radio pulsars 47

A, tsO2.58 days

i30~ 6.5B. t ~l.2.10yrs

onset of first stag.at mass exchange

13.0 6.5 P 2.58 days— — — — — — orb

C. I 1.2 • 10 yrs .210 yrs- ,~, I end ot fIrst stage

S .>. -sWt1(t’t~® I2 -- “ - ‘~~3a~1 I of mass exchange

170 / F°.’bo

Be star/

‘s~ 0. 1 - 1.48.lOyrsthe helium starhas exploded as

— ‘S.. ~‘ a supernova

I .. ,~) ~orb 30.53 days

Fig. 25. Conservative evolutionary scenario for the formation of a Be/X-ray binary out of a close pair of early B stars withmasses of 13.OM® and 6.5M®. The numbers indicate mass (in units of M®). After the end of the mass transfer the Be starpresumably has a circumstellar disk or shell ofmatter associated with the rapid rotation (induced by the previous accretionof matter with high angular momentum; from Habets [101]).

3.5.4. Formation of neutron stars in binary systems3.5.4.1. Outline ofthe formation processes of neutron stars in binaries. The neutron stars in X-ray

binaries can either have been formed by (i) the direct collapse of the nuclearly exhausted core ofan (initially) massive star, or (ii) by the accretion-induced collapse of a white dwarf in an oldbinary (a similar collapse might also be triggered by the coalescence of two white dwarfs in a veryclose white dwarf binary, cf. refs. [382,383,120]).The massive X-ray binaries have experienced the evolution of type (i). The reason why these

systems were not disrupted by the SN explosion that formed the neutron star is that at the momentof the explosion the star had already become the less massive component of the system, as aconsequence of a preceding stage of large-scale mass transfer [349,327,334]. As was shown byBlaauw [35], the system is disrupted only if more than half of the system mass is ejected in theexplosion, which is not the case here. Figure 25 depicts, as an example, the evolutionary history ofa typical B-emission X-ray binary [101]. The various evolutionary stages are described in the figurecaption. This type of close binary evolution, in which the mass transfer starts after the end of core-hydrogen burning, but before helium ignition, is called case B (in the terminology of Kippenhahnand Weigert [131], see section 3.5.3 and fig. I 9b). Basically the formation of the massive X-raybinaries can be understood in terms of so-called “conservative” models such as that depicted in fig.25, in which it is assumed that the total mass and total orbital angular momentum of the system

Conservativeevolution of aMassive binary

64 D. Bhattacharya and E.P.J. van den Heuvel, Binary and millisecond radio pulsars

ol FINAL EVOLUTION OF A CLOSE bI FINAL EVOLUTION OF A WIDEMASSIVE X-RAY BINARY MASSIVE B-emission) X-RAY BINARY

4~i.. OLD t UTRON (1) tot

OLD NEUTRON ~ ~ ~Oea~T~ 1 yeorSTAR • 1.4M0

(1) tot1 ~ ONSET OF16M0 14 N0 1 year ‘~ MASS TRANSFER

Mo12M.ONSETOF 2MASSTRANSFER ~

COMPLETE ao~~/ (2)

/ (2) tmt1.lOyr

SINGLE 3.0 •.1.4EUTRONSTAR~~ ‘TNORlc&ZYTK~ HeC,O/4 \ (3) t ~t,.1i-1Oyr d//~wifh/ — STAR (RED ‘ ........f P~U1RON to ci

ACCRETION DISK SUPERGIANT) t~l sie~

VERY STRONGSTELLAR WIND IfM~0 ~

(3) tet1n(1-5)imlO3yr Mco~

42Mo O)dE~>~0~~ 1.4 11 4SINGLE SECOND I I~f~trOfl neutr

9onl ~P youngNEUTRON STAR SPlR8~L-INI star I n utror,

• witk $ I (BINARY RADIO I s~OrWEAKMAGNETIC <1.4f~~.\1.6 ~- PULSAR) J starFIELD AND ht~”~ neutron~-~ —~(SYSTEM DISRUPTED)LOW SPACE dwarf

Fig. 32. The various possibilities for the final evolution of a High-Mass X-ray Binary. In all cases the onset of Roche-lobeoverflow leads to the formation of a common envelope and the occurrence of spiral-in. (a) In systems with orbital periodsless than about 1 year there is most probably not enough energy available in the orbit to eject the common envelope, andthe neutron star spirals down into the core of its companion. Subsequently, the envelope is ejected by the liberated accretionenergy flux, leaving a single recycled radio pulsar. (b) In systems with orbital periods larger than about a year the commonenvelope is ejected during spiral-in, and a close binary can be left, consisting of the neutron star and the core (consistingof helium and heavier elements) of the companion. Companions initially more massive than 8—12M® leave cores that willexplode as a supernova, leaving an eccentric-orbit binary pulsar or two runaway pulsars. Systems with companions lessthan massive than 8—l2Me leave close binaries with a circular orbit and a massive white dwarf companion, similar toP5R0655+64.

spun up to quite short rotation periods; for example, at L = LEdd a neutron star with a B field of2 x lO~~G can be spun up to 40 ms according to eq. (3.42). This explains the very short pulseperiod (59 ms) of the binary radio pulsar PSR1913+l6 [296], as well as that of X0538—66 (69ms). In view of its weak magnetic field the spin-down timescale of PSR1913+ 16 is enormous:P/P> 2 x 108 yr. On the other hand, the newborn neutron.star in this system is expected to havea strong field (B> 1012 G), implying a spin-down timescale < 5 x 106 yr. Thus, while the newbornpulsar in the system will have crossed the death line and have turned off within a few millionyears, the recycled one will remain observable for hundreds of millions of years, if its magneticfield does not decay significantly any more, as suggested by several authors [144,314,354,287,295].This explains in a natural way why in the 1913 + 16 system the recycled old pulsar is observed andthe newborn one is not [352].

High mass X-ray binary: final evolution

D. Bhattacharya and E.P.J. van den Heuvel, Binary andmillisecond radio pulsars 73

Helium core0.24 M0

~ •f o45x1l3~,,r~F’- 30d ~ 1.4

0.6 -- --

—6 I __~I

/~IlI ~. /

/-7 . . -.. ....-ci —~,,. /InItial ,,..~ _/~ lo6.0x10~,rjV~~A~.~‘

‘~ M20.9M0~ /~ ~117d WM~~’>. —8 \ 1.69 I

•~—g . 031 “~

~ final ,// t~6.1x1O

7yr—11 p,

117d •0 .1 .2 .3 .4 .5 .6 0.31 M0 1.69 N0

MC(MO I Helium white dwarf Neutron star

Fig. 34. Mass-transfer rate M for low-mass X-ray binaries Fig. 35. Evolution of a wide low-mass X-ray binary such asdriven by the evolutionary expansion of the mass donor Cygnus X-2 into a wide radio-pulsar binary with a circular(low-mass giant), as a function of its helium-core mass orbit and a low-mass helium white-dwarf companion, suchM~.The curves are for metallicity Z = 0.02. The dashed as PSR 1953+29. At the onset of the mass transfer, theline indicates the rates at the onset of the transfer, for a low-mass companion is a (sub-)giant with a degenerategiant mass of 0.9M@. The full line indicates the end of helium core of 0.24Mrn. Its light is generated by hydrogenthe mass transfer, and dotted lines indicate the evolution fusion in a shell around the core. The mass transfer fromof M during the transfer. An initial neutron-star mass of the giant to the neutron star is due to the slow expansion ofI .4Mg was adopted, and the mass transfer is assumed to the giant, driven by this hydrogen-shell burning. During thebe conservative (after Verbunt [3641). Figure 36 indicates mass transfer the orbit gradually expands (due to angular-the evolution of the orbital periods for the systems depicted momentum conservation) and after 8 x 108 yr the systemhere, terminates as a wide radio-pulsar binary (after Joss and

Rappapdrt [1291).

companion of mass 1 Me, Z = 0.02) in first approximation only on the initial orbital period P0,

roughly as:

(M) = —6 x 10’°(P0/l day) Me yr’. (3.60)

The mass-transfer lifetime of these systems is roughly:

t~i0.75Me/(M)~l.25x 109(lday/Po) yr. (3.61)

For other companion masses and chemical compositions, the relation becomes slightly different(see ref. [385]). Relations similar to eqs. (3.56) and (3.57) can also be derived for low mass starswith degenerate CO cores. These stars are on the asymptotic giant branch. They will occur as massdonors only in systems with initial orbital periods of a year or longer. In these systems very highmass-transfer rates (‘—i 1 0~Me yr1) are expected [138].

3.6.4.8. Origin of the wide binary radio pulsars with (nearly) circular orbits and low companionmasses. The orbital characteristics and companion masses of the wide radio-pulsar binaries show a

PH217: Aug-Dec 2003 7

more matter is taken away from the donor, degeneracy pressure beginsto become more important in supporting the star. Once the remainingconfiguration is almost fully degenerate the mass-radius relation reverses(star expands upon loss of mass) and the orbit begins to expand again. Suchsystems would pass through a minimum in orbital period (see fig. 4). Forinitial ⇠ 1M� components and solar composition, this minimum orbitalperiod works out to be ⇠ 80 minutes which is indeed observed amongbinary systems such as Cataclysmic Variables, consisting of White Dwarfaccretors and low-mass donors.

Mdonor

Po

rb

80

min

0.1 Msun

norm

al

degenera

te

Figure 4: Schematic evolution of the orbital period of a binary containing alow-mass donor and a compact star accretor. The mass transfer is sustainedby angular momentum loss. The orbital period passes through a minimumas the donor equation of state, originally thermal, turns degenerate causinga reversal of the mass-radius relation.

In binaries containing White Dwarf accretors the transferred mass is usu-ally not retained on the White Dwarf because of the following reason. Aftera period of accretion, at the base of the accreted layer the temperature and

Low-mass X-ray binary

Widesystems

Closesystems

10 h

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IUCAA-NCRA Graduate School 2016dipankar/gradschool2016/gradschool2016_stars.pdf · 1011 stars 1987A&A...184..104R Rana 1987. ... before the C-ignition temperature is reached and becomes