Archaic: very conveninient if

a) no calculations

b) all stellar objects are blackbodies 3 - 10000 K (i.e. stars)

c) use only photography in visual

But still in use

Take a star (alpha centuri): call it m=0

Take another star 100x fainter: call it m=5

Take another star 104xfainter: call it m=10

Algorithm for stars of same type as alpha centuri

1) Take number of times fainter I0/I

2) Take log10(I0/I)

3) Multiply by 2.5

exponential to the 2.51 power 2.51=

What happens is spectrus not the same

Three standard magnitudes

color excess

MU, MB, MV scaled for standard star

Could just quote flux densities at the 3 points

magnitudes very misleading is bulk of flux is outside optical

Sun 26.5 Faintest visible 6

Full Moon 13 Faintest Detectable 22

Sirius 1.5


Absolute Magnitude (M)

m=M at 10 pc

m at d

m at 10

but m = -log2.51 Id =

M = -log2.51 I10

M - m = -2.5 log Id + 2.5 log I10

= -2.5 log ()

= -2.5 log ()2

= -5 log d + 5 log 10

M = m - 5 log d + 5


Astrophysical Plasma

-Cosmic/Solar abundances

What does it emit in various circumstances

If we understand shysics then we can deduce circumstances of the matter

All photons are generated by the acceleration of charged particles

Will start with the simplest case:


Isolated charge


Larmor Formula

Relativistic Form:

(for colinear acceleration and velocity)








(for perpendicular acceleration and velocity)




We thus know the power here, but not the spectru. It gets complicates and E & M not appropriate here

Qualitative approach:

To become a photon electric field must be perturbed so that can add




then we can get a photon

if goes through many cycles then integral will be zero

Therefore, wt < 1 for photon creation





Conservation of energy predicts a result like this. You can’t put more energy into a photon than is lost from a particle.

What can cause acceleration of a particle?

Collisions with other particles

Bremsstrahlung Radiation

E & B Fields

E fields get shorted out

B fields Synchrotron and Cyclotron Radiation

Collisions between particles and particles spiralling in B fields account for virtually all orgination of photons


Optically thin photons escape as generated

Optically Grey or Thick photons interact with other matter before escaping

We will now discuss emission mechanisms Synchrotron and Bremsstrahlung in optically thin case


Cyclotron Radiation



Non-relativistic particle

Velocity = v

Mass = m


Therefore, cutoff at wc




With ensembles of particles: lines are not perfect d function


Widened by:

change in w (relativistic effect)

collisional broadening

self absorption

plasma distortion

non uniform B






How does a particle behave in such a field?

note: photon loses energy 106 times slower electrons are source

but let



loses energy on a timescale

/B2 = 10years/B2guess

Radiates at Hz = 2.8 B MHz




Synchrotron Radiation




Because of relativistic effects synchrotron does not radiate at fundamental frequency

In instantaneous rest from Larmor formula




Emits radiation at what particle thinks cyclotron frequency is, goes around at but due to time, emits at wc in its frame, but sees Brest = g Bobs emits at gwc









Main Output at




described exactly as modified Bessel function

Ensemble of Electrons

Power law distribution of energy

convolve functions

jn = emissivity per unit volume

Power law spectrum is seen



Synchrotron Decay Time


P= 1.6 * 10-15 B2g2 ergs/s

E= gmc2 =0.91*10–27 * 9*1020 g ergs

E=8.2*10-7 g ergs


solve for B & g




B ~ 5 gauss

g ~ 7