Homework #6

Question 1

(1. Accretion Physics)
What is definition of accretion efficiency?

Answer 1

n acc = Lacc / (dm/dt) c^2

Measures the fraction of rest mass energy converted to radiation during accretion:

Question 2

(1. Accretion Physics)
What physical energy source powers accretion luminosity?

Answer 2

Gravitational potential energy released as matter falls into a compact object.

Question 3

(1. (Accretion Physics - Energy released by Accretion)
What is gravitational potential energy released per unit mass during accretion?

Answer 3

E ~ Gm/R

Question 4

(1. (Accretion Physics - Energy released by Accretion)
Accretion efficiency expressed in terms of compact object properties.

Answer 4

n acc ~ GM/Rc^2

Question 5

(1. (Accretion Physics - Energy released by Accretion)
Why does accretion efficiency increase for compact objects?

Answer 5

Because gravitational potential energy increases as radius decreases so more energy is released per unit mass.

Question 6

(1. (Accretion Physics - Energy released by Accretion)

What physical quantity does GM/(Rc^2) represent?

Answer 6

Compactness parameter of an object.

Question 7

(1. (Accretion Physics - Nuclear vs. Accretion Energy)
Efficiency or nuclear fusion in stars.

Answer 7

n nuc ~ 0.007

Question 8

(1. (Accretion Physics - Nuclear vs. Accretion Energy)
Why can accretion onto compact objects be more efficient than nuclear fusion?

Answer 8

Because very deep gravitational potentials convert more mass energy into radiation.

Question 9

(1. (Accretion Physics - Nuclear vs. Accretion Energy)
How to compare types of efficiency?

Answer 9

n acc / n nuc

if ratio in decimals, accretion onto object less efficient than fusion.

if ratio is whole number, accretion is more efficient than fusion.

Question 10

(1. (Accretion Physics - Black Hold Accretion Radius)
Schwarzschild Radius

Answer 10

R s = 2GM/c^2

Critical radius of event horizon/boundary around black hole where light cannot escape.

Question 11

(1. (Accretion Physics - Black Hold Accretion Radius)
Why is inner radius of an accretion disk around a non-rotating black hole 3Rs?

Answer 11

Because stable circular orbits do not exist inside the radius. Matter inside this radius quickly plunges into the black hole.

Question 12

2. Rotational Stability of Compact Stars - Breakup Condition)

What determines whether a rotating star witll break apart.

Answer 12

The centrifugal force at the equator must not exceed gravity.

Question 13

2. Rotational Stability of Compact Stars - Breakup Condition)
   Condition for rotational stability at stellar surface.

Answer 13

mv^2/R less than or equal to GMm/R^2

so R less than or equal to Gm/v^2

Since v = 2piR/P, then R less than or equal to (GMP^2/4pi^2)^1/3

Question 14

2. Rotational Stability of Compact Stars - Breakup Condition)
   Rearrange stability condition to surface velocity.

Answer 14

v^2 less than or equal to GM/R

Question 15

2. Rotational Stability of Compact Stars - Breakup Condition)
   Physical interpretation of breakup condition.

Answer 15

The surface rotational velocity cannot exceed orbital velocity at the surface.

Question 16

2. Rotational Stability of Compact Stars - Spin Velocity)

Relation between surface rotational velocity and spin period

Answer 16

v = 2piR/P

Question 17

2. Rotational Stability of Compact Stars - Spin Velocity)
   What is surface velocity of a rotating star with period P?

Answer 17

v = 2piR/P

Question 18

2. Rotational Stability of Compact Stars - Spin Velocity)

Why can fast pulsars only be neutron stars?

Answer 18

Because large objects like white dwarves would exceed the breakup velocity if they rotated that fast.

Question 19

2. Rotational Stability of Compact Stars - Why Spin Period Constrains Object Type)

Why can spin period be used to identify compact objects?

Answer 19

Because white dwarfs, neutron stars, and black holes have very different radii which determines their maximum allowed spin rate.

Question 20

(3. X-Ray Binary Orbital Physics - Kepler’s 3rd Law)
Kepler’s third law for a binary system.

Answer 20

G(M1 + M2)/A^3 = (2pi/P)^2

Question 21

(3. X-Ray Binary Orbital Physics - Kepler’s 3rd Law)

What does A represent in binary orbital dynamics?

Answer 21

The total separation between the two stars.

A = r1 ( M2 + M1 / M2) via center of mass as:

M1r1 = M2r2

Question 22

(3. X-Ray Binary Orbital Physics - Kepler’s 3rd Law)

Center of Mass and why it must be true?

Answer 22

Since both stars orbit common center of mass.

Question 23

(3. X-Ray Binary Orbital Physics - Kepler’s 3rd Law)

How to derive mass function?

Answer 23

1. Start with Kepler’s 3rd Law.
2. Use A to rewrite interns of r1 and mass with center of mass formula.
3. Use velocity equation to find r1 as v = 2pir1/P.
4. Plug 3. Into 2.
5. Then plug this A into Kepler’s 3rd Law. Then isolate such that M’s on one side and rest are on the other. Add sini at end since we assume those are 1.

Question 24

(3. X-Ray Binary Orbital Physics - radial velocity observations)

What velocity do observers measure in binary system

Answer 24

Radial velocity component

Question 25

(3. X-Ray Binary Orbital Physics - radial velocity observations) Observed velocity relation

Answer 25

v obs = vsini i is inclination angle of orbit relative to observer

Question 26

(3. X-Ray Binary Orbital Physics - Binary Mass Function) What does the binary mass function allow astronomers to determine?

Answer 26

Lower limit on the mass of unseen companion.

Question 27

(3. X-Ray Binary Orbital Physics - Binary Mass Function) Why is the mass only a lower limit?

Answer 27

Because inclination is usually unknown and sini less than or equal to 1.

Question 28

Why are X-ray binaries associated with compact objects?

Answer 28

Because high accretion efficiency and high luminosities require deep gravitational potentials.

Question 29

Why do compact objects produce strong X-ray emission?

Answer 29

Accreting gas reaches very high temperature (10^7-10^8 K) when gravitational energy is converted to heat.

Question 30

(3. X-Ray Binary Orbital Physics - Pulsar Doppler Timing) What is non-relativistic Doppler shift formula for frequency?

Answer 30

change in v / v = vsini/c = vrad/c

Question 31

(3. X-Ray Binary Orbital Physics - Pulsar Doppler Timing) What velocity component contributes to Doppler shift in binary system?

Answer 31

Line of sight velocity vrad = vsini

Question 32

(3. X-Ray Binary Orbital Physics - Pulsar Doppler Timing) What is relationship between pulse and frequency

Answer 32

v = 1/P

Question 33

(3. X-Ray Binary Orbital Physics - Pulsar Doppler Timing) Relationship between fractional changes in frequency and period

Answer 33

change in v / v = - change in P/P Minus sign because inversely related

Question 34

(3. X-Ray Binary Orbital Physics - Pulsar Doppler Timing) Equation relating pulse period changes to orbital velocity

Answer 34

vsini/c = change in P / P Where change in P is observed pulsar period due to orbital motion.

Question 35

(3. X-Ray Binary Orbital Physics - Pulsar Doppler Timing) Why did orbital motion change the observed pulse period

Answer 35

Because Doppler shifts change arrival time of pulses as pulsar moves towards or away from observer.