synchrotron sources - national radio astronomy observatory€¦ · synchrotron radiation have...

Synchrotron SourcesSpectra of Optically Thin Sources

If a synchrotron source containing any arbitrary distribution of electron energies is optically thin( ), then its low-frequency spectrum is the superposition of the spectra from individualelectrons and can never rise more rapidly than the power of frequency. In other words,

the [negative] spectral index (be careful not to confuse this with the

electron pitch angle ) must always be greater than . Most astrophysical sources of

synchrotron radiation have spectral indices near at high frequencies where they areoptically thin, and as we shall soon see, their overall spectral indices primarily reflect theirelectron energy distributions.

The energy spectrum of cosmic-ray electrons in the local interstellar medium (Casadei, D., &Bindi, V. 2004, ApJ, 612,262). In the energy range above a few GeV, N(E) is a power lawwith slope .

Ü Ü 11=3

Ë d =d Ñ À logP· log · Ë Ë À1=3

Ë :7 Ù 0

Î :4 Ù 2

Synchrotron Sources http://www.cv.nrao.edu/course/astr534/SynchrotronSrcs...

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The observed energy distribution of cosmic-ray electrons in our Galaxy is roughly a power law:

where is the number of electrons per unit volume with energies to . The

energy range around is relevant to the production of radio radiation, and there thepower-law slope is . Because is nearly a power law over more than two

decades of energy and the critical frequency is proportional to energy squared, we expectthe synchrotron spectrum to reflect this power law over a frequency range of at least

. Consequently, we can ignore the detailed spectra of individual electrons, whichare smeared out in the observed spectrum by this broad power-law energy distribution. Wemake the very simple and crude approximation that each electron radiates all of its power

at the single frequency

which is very close to the critical frequency. Then the emission coefficient of synchrotronradiation by an ensemble of electrons is

where

Differentiating gives

so

N(E)dE E dE Ù K ÀÎ (5D1)

N(E)dE E E E + d

Í 0 Ø 1 4

Î 2:4 Ø + N(E) · c

(10 ) 0 2 2 = 1 4

P Û Ì Í cU = Àdt

dE=

3

4T

2 2B

· · Ù Í2G

Ï d· N(E)dE · = Àdt

dE

E m c m c : = Í e2 Ù

Ò·

·G

Ó1=2

e2

E

d·

dE Ù2·

G

1=2

m c ·e2 À1=2


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Eliminating in favor of and ignoring the physical constants in this equation for resultsin the proportionality

since . We finally get:

Since ,

Ë

That is, the synchrotron spectrum of a power-law energy distribution is itself a power law, andthe equation above relates the slopes of these two power laws.

Example: In our Galaxy , so we expect

and hence the (negative) spectral index should be

which is in agreement with observation. This is also the typical spectral index of most opticallythin extragalactic radio sources, even radio galaxies and quasars. It reflects the power-lawenergy distribution of cosmic rays accelerated in shocks, the shocks produced by supernovaremnants expanding into the ambient interstellar medium for example.

Ï Û Ì Í cU (KE ) · ÙÒ

3

4T

2 2B

ÓÀÎ

Ò

2·G

1=2

m c ·e2 À1=2Ó

E ·=· G Ï ·

Ï B (· ) · /Ò·

·G

Ó2

Ò·

·G

ÓÀÎ=2·G

À1=2

Ï B (· ) · /Ò·

B

Ó2

Ò·

B

ÓÀÎ=2B À1=2

· G / B

Ï · · / B(Î+1)=2 (1ÀÎ)=2 (5D2)

· ÀË / ·(1ÀÎ)=2

=2

Î À 1(5D3)

Î :4 Ù 2

Ï · · / B1:7 À0:7

Ë :7 ; Ù 0


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Synchrotron radiation (dot-dash line) from cosmic-ray electrons accelerated by the supernovaremnants of relatively massive ( ) and short-lived ( yr) stars dominates

the radio continuum emission of the nearby starburst galaxy M82 at frequencies GHz. Thermal emission (dashed line) from HII regions ionized primarily by even more massive( ) and shorter-lived stars is strongest between about 30 and 200 GHz. Atfrequencies well below 1 GHz, free-free absorption flattens the overall spectrum.

Minimum Energy and Equipartition

What is the minimum energy required to produce a synchrotron source of a given luminosity?The existence of the source requires relativistic electrons with some energy density and amagnetic field whose energy density is .

To estimate , we assume a power-law electron energy distribution

spanning the energy range to needed to produce synchrotron radiation over the

observed frequency range to . Then

M M > 8 Ì T 0 < 3Â 1 7

· 0 < 3

M 5M > 1 Ì

U eU =(8Ù) B = B2

U e

N(E) E Ù K ÀÎ

E min E max

· min · max

U N(E) E e =

Z

Emin

Emax

E d


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For a given synchrotron luminosity

Substituting and the synchrotron power emitted per electron

gives

Since electrons with energy emit most of the radiation seen at frequency , theelectron energy needed to produce radiation at frequency scales as

If we consider the energy content of only those electrons that emit in a fixed frequency range(e.g., from Hz to Hz), then the energy limits and are both

proportional to and

We conclude that

and we already know that

The "invisible" cosmic-ray protons and heavier ions emit negligible synchrotron power but they

L d·; =

Z

·min

·max

L·

L

Ue /N(E) E

REmin

Emax E d

À (dE=dt)N(E) EREmin

Emax d

N(E) E = K ÀÎ

(ÀdE=dt) E / B2 2

L

Ue /K EREmin

Emax E1ÀÎ d

KB E2REmin

Emax E2ÀÎ d

L

Ue /E j2ÀÎ

Emin

Emax

B E j2 3ÀÎEmin

Emax

E · B / E2

·

E / BÀ1=2

· 0 min Ø 1 7 · 0 max Ø 1 11 E min E max

B À1=2

L

Ue / (B )À1=2 2ÀÎ

B (B )2 À1=2 3ÀÎ =BÀ1+Î=2

B B2 À3=2+Î=2= BÀ3=2

U e / BÀ3=2 (5D4)

U : B / B2


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still contribute to the total cosmic-ray particle energy. If we call the ion/electron energy ratio ,then the total energy density in cosmic rays is . The total energy density of bothcosmic rays and magnetic fields is

U 1 )U

We cannot measure directly in distant radio sources, but cosmic rays collected near theEarth have .

The greatly differing dependences of and on means that the total (cosmic ray plus

magnetic) energy density has a fairly sharp minimum near the point at which

.

For a source of a given synchrotron luminosity, the particle energy density is

proportional to and the magnetic energy density is proportional to . The total

energy density has a fairly sharp minimum near equipartition of the particle and

magnetic energy densities ( ).

The minimum of the total energy density occurs at

Ñ (1 )U + Ñ e U

= ( + Ñ e + UB (5D5)

Ñ Ñ 0 Ù 4

U e U B B U(B)

(1 )U + Ñ e Ù UB

U 1 )U E Ñ ( + Ñ e

B À3=2 U B B 2

U = UE + UB

U E Ù UB

U


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First we evaluate the electron energy density.

so

Next we evaluate the magnetic-field energy density.

so

Inserting these results into the minimum-energy equation gives

At minimum energy, the ratio of particle to field energy is

This ratio is nearly unity. Thus minimum energy implies (near) equipartition of energy: thetotal cosmic-ray energy density (including the nonradiating ions) is nearly equal to

the total magnetic energy density . We don't really know if equipartition exists in mostsources, but radio astronomers often assume so, for several reasons:

(1) It is physically plausible—systems with interacting components often tend towardequipartition.

dU

dB=

dB

d[(1 )U ]+ Ñ e + UB = 0

B B dB

dUe Á UeÀ1 = À

Ò

2

3Ó

À5=2 3=2 = À 3

2B

dB

dUe= À

2B

3Ue

dB

dUB Á UBÀ1 =

B2

2B=

2

B

dB

dUB =B

2UB

dB

d[(1 )U ]+ Ñ e+

dB

dUB = 0 = À2B

3(1 )U+ Ñ e+

B

2UB

field energy

particle energy=

UB

(1 )U+ Ñ e=

3

4(5D6)

(1 )U + Ñ e

U B


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(2) Large and luminous extragalactic radio sources such as Cyg A have enormous energyrequirements even near equipartition; the problem of explaining the large energy is even worseotherwise.

(3) It eliminates an unknown parameter and permits estimates of the relativistic particleenergies and the magnetic field strengths of radio sources.

Getting the actual numerical values of the particle and magnetic field energies from thesynchrotron emission coefficient is a straightforward by tedious algebraic chore (see Rohlfs &Wilson Section 9.10). Below are the results (from Pacholczyk's Radio Astrophysics, p. 171). The functions and in these equations absorb the integrations from frequency to

and the physical constants in Gaussian cgs units.

For a spherical radio source with radius and magnetic field strength , the total magneticenergy is

E V :

The minimum-energy magnetic field is

B 4:5(1 )c L] R Gauss

and the corresponding energy in relativistic particles is

E (total) [(1 )L] R ergs

where the radio luminosity is conventionally integrated over the observable frequency range Hz to Hz,

is the source distance, is its flux density at frequency , and

The minimum total energy in relativistic particles and fields occurs when

;

c 12 c 13 · min

· max

R B

B = UB =8Ù

B2

3

4ÙR3

=6

B R2 3

min = [ + Ñ 122=7 À6=7 (5D7)

min = c13 + Ñ 4=7 9=7 (5D8)

L · 0 min = 1 7 · 0 max = 1 11

L d· ; ÙD S ; =

Z

·min

·max

L· L· = 4 2·

D S · ·

1 : + Ñ Ñenergy in relativistic electrons

energy in all relativistic particles

UB

(1 )U+ Ñ e Ù 4

1 + Î


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where

The synchrotron lifetime of a source is defined as the ratio of electron energy to the energyloss rate from synchrotron radiation:

Ü B

The functions and in Gaussian cgs units are plotted below.

Plots of as a function of [negative] spectral index for Hz

(dashed curves) and Hz (solid curves) and Hz, Hz, and Hz.

Î Ë :4 : = 2 + 1 Ø 2

L

Ü : ÑL

Ee

Ù c12 ?À3=2

(5D9)

c 12 c 13

c 12 Ë d =d Ñ À logS log · · 0 min = 1 6

10 7 · 0 max = 1 10 10 11 10 12


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Plots of as a function of [negative] spectral index for Hz

(dashed curves) and Hz (solid curves) and Hz, Hz, and Hz.

Example: What are the minimum-energy magnetic field strength and the minimum total energyof Cygnus A, a luminous double radio source (see the VLA image below) at distance Mpc (for km s Mpc )? The lobe radii are kpc and the total flux density ofCyg A is

First we convert the data from "astronomical" units to cgs units:

c 13 Ë d =d Ñ À logS log · · 0 min = 1 6

10 7 · 0 max = 1 10 10 11 10 12

D 30 Ù 2H 5 0 = 7 À1 À1 R 0 Ù 3

S 000 Jy · Ù 2

Ò·

GHz

ÓÀ0:8


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The spectral luminosity of Cyg A is

The total radio luminosity of Cyg A in the frequency range Hz to Hz is:

R 0 kpc :0 0 cm = 3 Âkpc

10 pc3

Âpc

3:09 0 cmÂ 1 18

Ù 9 Â 1 22

S 000 Jy · = 2

Ò

Jy

10 erg s Hz cmÀ23 À1 À1 À2ÓÒ ·

10 Hz9

ÓÀ0:8

S · =cm2

3:17 0 erg s HzÂ 1 À13 À1 À1

ÂÒ·

Hz

ÓÀ0:8

D 30 Mpc :1 0 cm = 2 ÂMpc

10 pc6

Âpc

3:09 0 cmÂ 1 18

Ù 7 Â 1 26

L ÙD S Ù(7:1 0 cm) · Ù 4 2· = 4 Â 1 26 2 Â

cm2

3:17 0 erg s HzÂ 1 À13 À1 À1

ÂÒ·

Hz

ÓÀ0:8

L :0 0 erg s Hz · Ù 2 Â 1 42 À1 À1 ÂÒ·

Hz

ÓÀ0:8

10 7 10 11

L d· =

Z

10 Hz7

10 Hz11

L·

L :0 0 erg s Hz Ù 2 Â 1 42 À1 À1

Ò

0:2

·0:2Ó Ì Ì Ì Ì 10 Hz7

10 Hz11

L :0 0 erg s Ù 2 Â 1 42 À1

Ô

0:2

(10 ) 10 )11 0:2 À ( 7 0:2 Õ

L :33 0 erg s Ù 1 Â 1 45 À1


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In units of the bolometric (jargon for "as observed by a bolometer" and meaning integratedover all frequencies) solar luminosity erg s , the radio luminosity of Cyg Ais

:5 0

Thus the radio power from Cyg A exceeds the bolometric output from a galaxy of stars similarto our Galaxy. The energy appears to originate in a compact object at the center of the hostgalaxy. How massive must this compact object be to produce such a luminous source?

Eddington Limit

What is the maximum luminosity of an astronomical object of total mass in a steady state?The outward radiation pressure cannot exceed gravity. For example, radiation pressure wouldexpel the outer layers of a star in the form of a wind, or accretion onto a compact objectwould be disrupted. Even if the atmosphere or infalling material is ionized hydrogen, the freeelectrons will Thomson-scatter outflowing radiation. Each electron being blown away byradiation pressure will drag along one proton ( ) to maintain charge neutrality.

Balancing the radiation and gravitational forces on each electron/proton pair at distance fromthe accreting object gives the Eddington Luminosity:

Note that the distance drops out and

In cgs units

L (erg s )

Normalized to "solar" units erg s and g,

L :83 0 Ì Ù 3 Â 1 33 À1

L

LÌÙ

3:83 0 erg sÂ 1 33 À1

1:33 0 erg sÂ 1 45 À1

Ù 3 Â 1 11

M

m p µ me

r

Û LE

4Ùr c2 T =r2

GM(m )p +me Ùr2

GMmp

r

L E Ù ÛT

4ÙGMm cp

EÀ1 =

6:65 0 cmÂ 1 À25 2

4Ù :67 0 dyne cm g :66 0 g 0 cm sÂ 6 Â 1 À8 2 À2 ÂM Â 1 Â 1 À24 Â 3Â 1 10 À1

L (erg s ) :28 0 (g) EÀ1 = 6 Â 1 4 M

L :83 0 Ì Ù 3 Â 1 33 À1 M :99 0 Ì Ù 1 Â 1 33


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As the mass of a main-sequence star approaches , its luminosity approaches itsEddington luminosity. Very massive stars often have radiation-driven winds, and stable starsmore massive than may not be possible.

Example: What does the Eddington limit give for the minimum mass for a source of luminosity? We make the assumption that the average luminosity of the central mass

was at least this much at some times, so

Note that the Eddington mass limit depends only on the instantaneous power emitted by thesource, not on the total energy of the source, the source age, or any other indicator of itshistory.

Temperatures of Eddington-limited accretion disks near black holes

The Eddington limit is directly applicable to quasars and other sources having having luminousaccretion disks. If the gravitational energy released by accretion is thermalized, the hottest andhence brightest material will be concentrated just outside the innermost stable orbitsurrounding the central rotating black hole. The radius of this orbit is

where is the gravitational radius or Schwarzschild radius. If the compactobject is accreting enough matter to approach its Eddington luminosity, the combination ofluminosity and radius determines the blackbody temperature of the inner accretion disk.

ÒLE

LÌ

ÓÙ

3:83 0 erg sÂ 1 33 À1

6:28 0 :99 0 gÂ 1 4 Â 1 Â 1 33 ÒM

MÌ

Ó

:3 0 ÒLE

LÌ

ÓÙ 3 Â 1 4

ÒM

MÌ

Ó(5D10)

M 00M Ù 1 Ì

100M Ì

L :5 0 L Ù 3 Â 1 11Ì

Õ 0 ÒM

MÌ

Ó

3:3 0Â 1 4

3:5 0Â 1 11

Ù 1 7

r r ; = 3 g = 3Âc2

2GM

r GM=c g = 2 2

T

L Ùr ÛT Ù 4 2 4 Ù LE


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The more massive the black hole, the cooler. Inserting cgs values for the constants gives

In units of g,

Example: The spectrum of the quasar 3C 273 is the superposition of a power-law fromsynchrotron radiation and a thermal "big blue bump" peaking at ultraviolet wavelengths.

4Ù ÛT Ù Ò

c26GM

Ó24 Ù 4

Ò

ÛT

GMm cpÓ

T M 4 ÙÒ

m cp5

36GÛÛT

ÓÀ1

T M 4 ÙÒ

1:66 0 g 3 0 cm s )Â 1 À24 ( Â 1 10 À1 5

36 :67 0 erg cm :65 0 cmÂ 6 Â 1 À8 À2 sÀ1 KÀ4 Â 6 Â 1 À25 2

ÓÀ1

T :46 0 g K 4 Ù 4 Â 1 62 4 MÀ1

M :99 0 Ì Ù 1 Â 1 33

:2 0 ÒT

K

ÓÙ 2 Â 1 7

Ò

M

MÌÓ1=4


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The spectrum of 3C273 (Malkom, M. A., & Sargent, W. L. W. 1982, ApJ, 254, 22).

If then K, in agreement with the observed thermal spectrum andaccounting for the strong emission-line spectrum of ionized hydrogen. Many quasars havesimilar blue bumps and appear to be accreting at rates approaching the Eddington limit. Notethat black holes with masses of only a few accreting at the Eddington limit will have much

higher temperatures K and be strong thermal X-ray sources.

Returning to Cyg A, we estimate the magnetic field strength that minimizes the totalenergy in relativistic particles and magnetic fields. We approximate Cyg A by two equal lobes ofradius kpc and luminosity , where is the radio luminosity of the whole source.

The value of is poorly constrained in extragalactic radio sources such as Cyg A. The cosmicrays accelerated by a supermassive black hole might be primarily electrons and positrons.Electrons and positrons have the same mass and charge (except for sign), so they are equallyefficient at emitting synchrotron radiation and . If electrons and protons are acclerated tothe same velocities (same ), then the protons carry as much energy but

M 0 M Ø 1 9Ì T 0 Ø 1 5

M ÌT 0 Ø 1 7

B min

R 0 Ù 3 L=2 L

B 4:5(1 )c (L=2)] R min Ù [ + Ñ 122=7 À6=7

B 4:5 :9 0 :33 0 erg s =2) (9 0 cm) (1 ) min Ù ( Â 3 Â 1 7 Â 1 Â 1 45 À1 2=7 Â 1 22 À6=7 + Ñ 2=7

Ñ

Ñ Ù 1Í m =m 0 p e Ø 2Â 1 3


15 of 22 10/21/2008 10:50 AM

emit almost nothing and . Fortunately, is only weakly dependent on .

Varying from 1 to only changes from about 1 to 9.

The minimum total energy of Cyg A is twice the energy of each lobe:

where is in the range of about 1 to 80.

This enormous energy can be used to set another lower limit to the mass of the central objectpowering the radio source. If mass could be converted to energy with 100% efficiency, theminimum mass needed to produce would be

M 0 g

This is a very conservative lower limit. Nuclear fusion can only convert mass to energy withabout 1% efficiency, so if the energy source were nuclear fusion. Accretion

onto a spinning black hole can result in efficiencies up to in theory, so it is

consistent with . In the literature, it is often assumed that mass is converted to

energy with about 10% efficiency; this yields . The small size of the radio coreimplied by Very Long Baseline Interferometery (VLBI) and variability of the core flux on timescales of months to years combined with the large minimum masses estimated from the

Ñ 0 Ø 2Â 1 3 B min / Ñ2=7 Ñ Ñ 2 0 Â 1 3 (1 ) + Ñ 2=7

B :45 0 :1 0 min Ù 1 Â 1 15 Â 2 Â 1 À20 Â (1 to 9) Gauss

B 30 to 300) 0 0 Gauss min Ù ( Â 1 À6 Ø 1 À4

E (lobes) [(1 )L] R min Ù 2 Â c13 + Ñ 4=7 9=7

E :0 0 (9 0 cm) 1 ) ; min Ù 2Â 2 Â 1 4

Ò

2

1:33 0 erg sÂ 1 45 À1Ó4=7

Â 1 22 9=7 Â ( + Ñ 4=7

(1 ) + Ñ 4=7

E 0 :1 0 :26 0 1 to 80) ergs min Ù 4Â 1 4 Â 4 Â 1 25 Â 3 Â 1 29 Â (

E :4 0 1 to 80) ergs 0 ergs min Ù 5 Â 1 59 Â ( Ø 5Â 1 60

E min

Õc2Emin Ù 5 0 ergsÂ 1 60

(3 0 cm s )Â 1 10 À1 2Ù 6Â 1 39

M 0 g 0 M Õ 6Â 1 39

ÒMÌ

1:99 0 gÂ 1 33

ÓÙ 3Â 1 6

Ì

M 0 M > 3Â 1 8Ì

(1 ) :4 À 3À1=2 Ù 0

M 0 M > 1 7Ì

M 0 M > 3Â 1 7Ì


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Eddington limit and the total energy of the radio lobes together imply that the compact,massive object powering the radio source is a supermassive black hole. The adjectivesupermassive is used to indicate black holes much more massive than the most massivestars, .

A lower limit to the age of the radio source Cyg A is the synchrotron lifetime of the relativisticelectrons estimated by taking the ratio of the electron energy to the observed synchrotronluminosity:

Ü 0 s 0 s 0 yr

Since each electron radiates energy at a rate proportional to and the critical frequency isproportional to , the most energetic electrons emitting at the highest frequencies have theshortest lifetimes. The rapid depletion of high-energy electrons causes the emitted radiospectrum to steepen at high frequencies.

Ø 00M 1 Ì

ÕL

E =(1 )min + Ñ Ù 5:4 0 erg 1 )Â 1 59 ( + Ñ 4=7

1:33 0 erg s 1 )Â 1 45 À1 ( + ÑÙ 4Â 1 14 Â ÑÀ3=7 Ø 1 14 Ø 3Â 1 6

E 2

E 2


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The radio spectrum of Cyg A (and Cas A, Vir A) from Baars, J. W. M. et al. 1977, A&A, 61, 99. Note the spectral steepening above MHz.

Suppose that new relativistic electrons are continously injected with a power-law energydistribution into a radio source. After a long time, electrons emitting at

frequencies higher than will have been depleted by radiative losses , so thesehigh-energy electrons will eventually have an energy distribution .

Consequently, the (negative) spectral index will be at low frequencies and

approach at higher frequencies; the high-frequency

spectrum steepens by .

If the observed cutoff frequency is very high, the synchrotron lifetime of electrons with may be less than the time needed for new relativistic electrons to travel from the core

to the emitting feature in a jet or lobe. This implies in situ acceleration—something outside thecore (e.g., shocks in the jet) must replenish the supply of relativistic electrons.

Optical synchrotron emission in the radio jet of Virgo A = M87. Image credit

· 0 Ø 1 3

N(E) / EÀÎ0

· / E2

N(E) / EÀ(Î +1)0

Ë Î )=2 0 = ( 0 À 1

Ë Î )=2 Ë =2) = ( 0 + 1À 1 = ( 0 + 1

ÁË =2 = 1

· · c Ø ·


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Synchrotron Self-Absorption

For every emission process there is an associated absorption process. The emitting particles ina source in local thermodynamic equilibrium (LTE) have a Maxwellian energy distribution, andsuch a source is called a thermal source. If the particle temperature is , the source cannothave a brightness temperature greater than . If the energy distribution of relativistic electronsin a synchrotron source were a (relativistic) Maxwellian, those electrons would have acharacteristic temperature , and synchrotron self-absorption would prevent the

brightness temperature from exceeding . Astrophysical synchrotron sources are often callednonthermal sources because the energy distribution of the relativistic electrons is a powerlaw and there is no single electron temperature . However, self-absorption occurs regardlessof the energy distribution.

In the approximation that electrons with energy in a magnetic field of strength emit only at the critical frequency

the Lorentz factor of electrons emitting at frequency is:

Since only electrons of one particular energy contribute to the emission and absorption at anyone frequency in that approximation, the other electrons could have a relativistic Maxwellianenergy distribution to match without changing the resulting emission and absorption at thatfrequency. Thus we expect that a sufficiently bright synchrotron source will be optically thick,and the brightness temperature at any frequency cannot exceed the effective temperature ofthose electrons emitting at that frequency.

In an ultrarelativistic gas, the ratio of specific heats at constant pressure and at constantvolume is , not the nonrelativistic , so the relation between electron energy

and temperature is

Thus the effective temperature of a relativistic electron is

T

Eliminating in favor of gives the effective temperature of those electrons accounting for

T T

T =3k Ø E

T

T

E m c = Í e2 B

· ; c ØÍ eB2

2Ùm ce

Í ·

Í : ÙÒ

eB

2Ùm c·eÓ1=2

c =c =3 p v = 4 5=3 E T e

E kT ; not (3=2)kT : = 3 e e

e =E

3k=

3k

Ím ce2

Í ·


19 of 22 10/21/2008 10:50 AM

most of the radiation at frequency :

Numerically,

br>

Example: What is the effective temperature of the relativistic electrons emitting synchrotronradiation at Hz if Gauss Gauss?

At a sufficiently low frequency , the brightness temperature of any synchrotron source will

approach the effective electron temperature at that frequency and the source will becomeopaque. Starting with the definition of :

I

and setting gives

Then, because flux density is proportional to for a source subtending a given solid angle ,the spectrum of a synchrotron self-absorbed and spatially homogeneous source is a powerlaw of slope :

independent of the slope of the electron-energy spectrum. The flux density of an opaque buttruly thermal source (e.g., an HII region) is proportional to ; the extra for synchrotronradiation comes from the fact that .

·

T e ÙÒ

eB

2Ùm c·eÓ1=2

3k

m ce2

:18 0 Ò

K

TeÓÙ 1 Â 1 6

Ò·

Hz

Ó1=2ÒB

Gauss

ÓÀ1=2

(5D11)

· :1 GHz 0 = 0 = 1 8 B 00 = 1 Ö = 0 1 À4

:18 0 10 ) (10 ) 0 Ò

K

TeÓÙ 1 Â 1 6 Â ( 8 1=2 À4 À1=2 Ù 1 12

· T b

T eT b

· =c2

2kT ·b2

T b Ù Te

I · B · Ùc2

2kT ·e2

/ ·1=2 2 À1=2

I · Ê

5=2

S(·) / ·5=2 (5D12)

Î · 2 · 1=2

T e / ·1=2


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The spectrum of a homogeneous cylindrical synchrotron source is a power law with slope at low frequencies where . Astrophysical sources are inhomogeneous, so

their actual low-frequency spectral slopes are smaller than 5/2. The optical depth of thesource is at .

We can invert the equation for to estimate the magnetic field strength in a self-absorbedsource whose brightness temperature has been measured.

Example: If a self-absorbed radio source is observed to have K at GHz, themagnetic field strength is

ÀË =2 = 5 Ü µ 1

Ü = 1 · = ·1

T e

:4 0 Ò

B

Gauss

ÓÙ 1 Â 1 12

Ò·

Hz

ÓÒ

K

Tb

ÓÀ2

(5D13)

T 0 b Ù 1 11 · = 1

:4 0 0 10 ) :1 Ò

B

Gauss

ÓÙ 1 Â 1 12 Â 1 9 Â ( 11 À2 Ù 0


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The spectra of "real" radio sources reflect the idealized spectra of uniform sources, but theyare more complex because real sources have nonuniform magnetic fields and electron energydistributions in geometrically complex structures. Representative spectra of powerful radiogalaxies and quasars are illustrated below.

Spectra of radio galaxies and quasars. The radio source 3C 84 in the nearby galaxy NGC 1275contains a very compact nuclear component that is opaque below about 20 GHz. The radiogalaxy 3C 123 is transparent at all plotted frequencies, and its spectrum steepens above afew GHz. The quasar 3C 48 is synchrotron self-absorbed only below 100 MHz, while thequasar 3C 454.3 contains structures that become opaque at widely differing frequences. [Kellermann, K. I., & Owen, F. N. 1988, in Galactic and Extragalactic Radio Astronomy, eds. G.L. Verschuur & K. I. Kellermann (Springer Verlag)]


22 of 22 10/21/2008 10:50 AM

synchrotron sources - national radio astronomy observatory€¦ · synchrotron radiation have...

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