Homework #5

Question 1

What emission mechanism produces X-ray radiation in the Crab Nebula?

Answer 1

Synchrotron radiation from relativistic electrons in magnetic field.

Question 2

What observational signature suggests synchrotron radiation in astrophysical sources?

Answer 2

Power-law spectrum

Question 3

Why does synchrotron emission produce a power-law photon spectrum?

Answer 3

The electron energy distribution is a power law and synchrotron emission maps that into a power-law photon spectrum.

Question 4

Physical conditions required for synchrotron radiation?

Answer 4

1. Relativistic electrons
2. Magnetic field
3. Electrons spiraling into the field

Question 5

Characteristic synchrotron frequency of electron

Answer 5

v ~ 3gamma^2 eB/4pimec sin theta

where theta I pitch angle

Question 6

Key scaling of synchrotron emission frequency

Answer 6

v ~ gamma^2B

higher energy electrons emit higher frequency radiation

Question 7

What is gamma

Answer 7

Lorentz factor: E/mec^2

Question 8

Relation between photon energy and frequency

Answer 8

E = hv

Question 9

Typical X-ray frequency?

Answer 9

v ~ 10^17 - 10^18 Hz

Question 10

Power radiated by relativistic electron via synchrotron radiation

Answer 10

P = 4/3 sigma_T c gamma^2 U_B

Question 11

Magnetic field energy density

Answer 11

U_B = B^2/8pi

Question 12

Synchrotron Cooling Time

Answer 12

tcool = E/P

Question 13

Simplified Synchrotron cooling time formula

Answer 13

tcool = 1/ gammabetta^2 B^2sin^theta

Question 14

Scaling of synchrotron cooling time

Answer 14

tcool ~ 1/B^2 gamma

High energy electrons cool faster.

Question 15

What does synchrotron cooling time represent

Answer 15

Time for electron to lose significant fraction of its energy due to synchrotron radiation.

Question 16

Angular frequency in terms of pulse period

Answer 16

w ~ 2pi/P

Question 17

Rotational kinetic energy of spinning object

Answer 17

W = 1/2 Iw^2

Question 18

Rate of rotational energy loss

Answer 18

dW/dt = Iw dw/dt

Question 19

Relation between dw/dt and dP/dt

Answer 19

dw/dt = - 2pi/P^2 dP/dt

Question 20

Why must relativistic electrons in Crab Nebula be continuously accelerating?

Answer 20

Synchrotron cooling times are much shorter than nebula age so the electrons lose energy quickly.

Question 21

What is likely energy source powering Crab Nebula synchrotron emission?

Answer 21

Spin-down energy of Crab pulsar

Question 22

What physical process powers pulsar spin-down luminosity?

Answer 22

Loss of rotational kinetic energy

Question 23

Why do high energy electrons emit higher frequency synchrotron radiation?

Answer 23

Because

v ~ gamma^2

so radiation frequency increased rapidly with electron energy.

Question 24

Lorentz factor from synchrotron radiation

Answer 24

Gamma = ( 4pimecv/3eBsintheta)^1/2

Question 25

Finger kotentz factor to electron energy

Answer 25

E = gamma mec^2