Compton Scattering II

Compton Scattering II 1 Introduction In the previous chapter we considered the total power produced by a single electron from inverse Compton scattering. This is useful but limited information. Here we calculate the resulting spectra. 1.1 Importance This emission mechanism is frequently encountered in a number of areas, for example: The X-ray emission from the lobes of radio galaxies can High Energy Astrophysics: Compton Scattering II 1/94

be used, along with the synchrotron emission to estimate the magnetic field and the energy density of particles and magnetic field. The additional information available from X-ray observations unlocks the degeneracy between particle estimates and fields involved in the expression for, say, the surface brightness of synchrotron emission. Such estimates can then be compared with the minimum energy estimates obtained from synchrotron theory. Blazars emit X-ray synchrotron and GeV or TeV -rays. The simultaneous measurement of these spectra enable us to estimate the properties of blazar jets, even though they are unresolved. It is well to keep in mind two possible scenarios: High Energy Astrophysics: Compton Scattering II 2/94

1.The soft photons which are scattered originate from an external source, e.g. starlight or the microwave background. This is known as External Inverse Compton (EIC) emission. 2.The soft photons originate from within the source itself and are scattered by the same population of relativistic electrons. This is known as Synchrotron Self Compton (SSC) emission. High Energy Astrophysics: Compton Scattering II 3/94

1.2 Example The western hot spots in Cygnus A. The radio image is shown as a greyscale; the X-ray image is shown as contours overlaid on the radio image. From Wilson et al. (2000) astro-ph 0009308 High Energy Astrophysics: Compton Scattering II 4/94

2 Inverse compton spectrum from a single relativistic electron 2.1 Relative importance of SSC and EIC emission One can readily ascertain the relative importance of SSC and EIC emission from the expression for the Compton power: 4 P Compton -- 3 T c 2 2 U ph 4 -- 3 T c 2 U ph for relativistic particles (1) High Energy Astrophysics: Compton Scattering II 5/94

Since the power is proportional to the radiation energy density, SSC is going to be more important than EIC when Radiation energy density of the synchrotron emission Radiation energy density of background emission (2) High Energy Astrophysics: Compton Scattering II 6/94

2.2 Calculation of the emitted spectrum Lab frame Electron rest frame 1 1 h 1 v 1 S h 1 S Electron scattering in laboratory frame and rest frame of electron Note that all angles are measured clockwise from the positive x-axis as defined by the electron velocity. High Energy Astrophysics: Compton Scattering II 7/94

Outline of what happens in practice Incident photons v 2 v Lab frame Scattered photons Electron rest frame High Energy Astrophysics: Compton Scattering II 8/94

Photon-electron collisions in the rest and lab frames One of the simplifications of inverse compton emission from relativistic electrons is due to the aberration of the photons in the rest frame of the electron. Let cos for the incoming photon in the rest frame and cos for the incoming photon in the lab frame. Then the relativistic aberration formula gives: --------------- 1 (3) High Energy Astrophysics: Compton Scattering II 9/94

For 1and most values of : The only exception is when and then 1 ----------- 1 1 1 1 1 Thus the values of cluster around 1 for high Lorentz factors. This corresponds to virtually head on collisions between electrons and photons in the rest frame. The only exception is for collisions which are close to overtaking. High Energy Astrophysics: Compton Scattering II 10/94 (4) (5) (6)

μ (electron rest frame) 1.0 0.5 0.0-0.5 γ 10-1.0-1.0-0.5 0.0 0.5 1.0 μ (lab frame) γ 2 The plot at the left shows the result of the aberration formula for cos, for 2 different values of. It is well to remember that the lab frame distribution is isotropic so that 1 is going to be the most common result. High Energy Astrophysics: Compton Scattering II 11/94

Electron energy On the other hand, the ratio of the photon energy in the rest frame to that in the lab frame is given by Rest frame photon energy ------------------------------------------------------------- Lab frame photon energy and varies linearly with. There is no specific value of which is favoured. --- 1 (7) --- High Energy Astrophysics: Compton Scattering II 12/94

Minimum and maximum values of rest frame photon energy The minimum value is for 1 1 1 1 1 + -------- 2 2 ----- 2 (8) The maximum value is at 1 1 + 2 (9) High Energy Astrophysics: Compton Scattering II 13/94

That is, ----- 2 2 (10) 2.3 Evaluation of rate of scattering We consider the situation S 1 first in the rest frame. The complexity of the scattering 1 is considerably simpli- fied by the fact that the photons are effectively in- High Energy Astrophysics: Compton Scattering II 14/94

cident on the electron at one angle. This means that we only have to consider one scattering angle for the scattered photons. In the rest frame: dn ------------------- dtd 1 No of photons scattered per unit time per unit solid angle in the electron rest frame p d T ----------- cfp d 1 d 3 p (11) High Energy Astrophysics: Compton Scattering II 15/94

The differential cross section is given by Thomson scattering: d T ----------- d 1 r 0 2 ---- 1 + 2 1 2 (12) We can write the cross-section this way because we are approximating the photons as all coming in along the x-axis. High Energy Astrophysics: Compton Scattering II 16/94

Relation to distribution in lab frame As we did for the total power calculation, we transform the integral over the rest frame momentum space into one over the lab frame momentum space. Note that d 3 p p 2 sindpdd p 2 dpdcosd p 2 dpdd so that integrations over (13) become integrations over. Also sin d d d 0 1 1 1 1 (14) High Energy Astrophysics: Compton Scattering II 17/94

Hence, d 3 p p 2 dpdd in integrations over momentum space. In Compton Scattering I, we found that: d 3 p 1 d 3 p (15) (16) Hence, fpd 3 p fp1 d 3 p (17) High Energy Astrophysics: Compton Scattering II 18/94

We also have 1 (18) Since the distribution in the lab frame is isotropic. Therefore, 4fpp 2 dp nd (19) And fpd 3 p --- fpd 3 p --- fp p 2 dpdd 1 ----- --- nddd 4 (20) (21) High Energy Astrophysics: Compton Scattering II 19/94

The integral over is easy: dn ------------------- dtd 1 1 ----- 4 1 -- 2 p d T ----------- c --- d 1 nddd d T ----------- c --- ndd d 1 (22) High Energy Astrophysics: Compton Scattering II 20/94

Change of variables from to We now make a change of variables from to utilising the transformation 1 (23) That is we are replacing by the photon energy seen by the electron in the rest frame. Our reason for doing this is that, later on, we relate to the energy of the scattered photon and thus derive the differential no of photons scattered per unit energy of the final scattered energy that is, the energy we receive at the telescope. High Energy Astrophysics: Compton Scattering II 21/94

Remember: Energy of incident photon in lab frame Energy of incident photon in rest frame (24) The Jacobean ----------------- dd 1 0 1 1 --------dd 1 ----dd (25) High Energy Astrophysics: Compton Scattering II 22/94

Hence, dn ------------------- dtd 1 1 -- 2 We now write this in the form: d T ----------- c --- ndd d 1 d T ----------- c d 1 dn --------------------------------- c d T ----------- dtd 1 dd d 1 ----------- ndd 2 2 ----------- n 2 2 (26) (27) (28) High Energy Astrophysics: Compton Scattering II 23/94

Scattering in the Thomson limit Now use the Thomson limit approximation for the energy of the scattered photon: 1 (29) As we saw in Compton Scattering I, the condition for this is that «m e c 2 (30) High Energy Astrophysics: Compton Scattering II 24/94

Using 1 gives: dn ------------------------------------ c d T ----------- dtd 1 dd 1 d 1 cr 0 2 1 ----------- 2 2 n ------- 1 + 2 1 2 1 ----------- 2 2 n (31) High Energy Astrophysics: Compton Scattering II 25/94

Relationship to the scattering rate in the lab frame The next step is to relate this differential expression to quantities in the lab frame. The two main variables to transform are t and 1. We use the Lorentz transformation for time t t x + ------- c (32) We also use the Lorentz transformation relating photon energy in the lab frame to photon energy in the rest frame given that the scattered photon has a direction given by 1 cos 1. High Energy Astrophysics: Compton Scattering II 26/94

This is: 1 1 1 + 1 (33) These transformations imply that: dt dt d 1 1 + 1 d 1 dt 1 dt d 1 d 1 ---------------------------- 1 + 1 (34) High Energy Astrophysics: Compton Scattering II 27/94

Also, since dn dn -------------------------------- dtd 1 dd 1 is a number it is a Lorentz invariant and dn ------------------------------------ dtd 1 dd 1 dt ------ dt d 1 --------- d 1 cr2 0 1 + 1 2 1 n ------- 2 ---------------------------- 1 + 1 2 ----------- 4 2 ---------- (35) High Energy Astrophysics: Compton Scattering II 28/94

Rate of scattering in all directions We now calculate the rate of scattering of photons into all directions. We integrate over solid angle in the rest frame, with d 1 d 1 d dn ------------------- dtdd 1 c r 0 2 1 n ---- 2 ----------- 4 2 ---------- 2 0 1max 1 min 1 + 1 2 ------------------------ 1 + 1 2 d 1 d (36) High Energy Astrophysics: Compton Scattering II 29/94

Limits on scattered energy in lab frame In order to complete the above integral, we have to determine the limits on 1. We first calculate the limits on 1. We have the relation between rest frame incident soft photon energy and lab frame scattered energy 1 : 1 1 + 1 1 + 1 and the previously derived limits on : ----- 2 2 (37) (38) High Energy Astrophysics: Compton Scattering II 30/94

For a given the upper limit on 1 is obtained for 1 1. This gives 1 2 Therefore, from the upper limit on 1 4 2 (39) That is, the kinematics of Compton scattering establishes a maximum energy, 4 2 of the scattered photons. High Energy Astrophysics: Compton Scattering II 31/94

Later we make use of the variable: Final photon energy x ------------------------------------------------------------ Maximum photon energy Limits on 1 for a fixed 1 and Since 1 ----------- 4 2 (40) 1 1 + 1 2 2 1 + 1 (41) High Energy Astrophysics: Compton Scattering II 32/94

then, for a given 1 1 + 1 This provides the lower limit on the integral over 1. 1 ----------- 2 2 (42) We also have from the lower limit on : 1 1 + 1 ----- 1 + 2 1 --1 + 2 1 (43) High Energy Astrophysics: Compton Scattering II 33/94

Therefore, We also have the obvious constraint on so that 1 + 1 2 1 ---- 1 + 1 2 (44) 1, namely 1 1 (45) Since we are normally interested in limit is usually the most relevant. 1» 1 then the latter High Energy Astrophysics: Compton Scattering II 34/94

Summary of limits i.e. and 1 ----------- 2 2 1 + 1 2 1 1 min ----------- 2 2 1 1max 1 (46) (47) (48) High Energy Astrophysics: Compton Scattering II 35/94

Evaluation of integral The indefinite integral 1 + 1 2 ------------------------ 1 + 1 2 d 1 2 1 ---------------- 2ln1 + 1 + 1 1 (49) Evaluation between the lower and upper limits gives: 1 1 2 1 ----------- 4 2 + 2ln----------- 4 2 + 1 1 ----------- 4 2 (50) High Energy Astrophysics: Compton Scattering II 36/94

Putting it all together: dn ------------------- c r 0 2 ---- dtdd 1 2 1 n ----------- 4 2 ---------- 2 0 1max 1 min 1 + 1 2 ------------------------ 1 + 1 2 d 1 d 2cr2 -------------- 0 n ---------- 2 (51) 1 1 1 1 + ----------- 4 2 + 2 ----------- 4 2 ln----------- 4 2 1 2 2----------- 4 2 High Energy Astrophysics: Compton Scattering II 37/94

We put x 1 ----------- 4 2 r2 3 0 -- 8 T (52) to obtain dn ------------------- dtdd 1 3 -- c T 4 ------------- n ---------- F C x 2 F C x 2xlnx+ x + 1 2x 2 (53) High Energy Astrophysics: Compton Scattering II 38/94

F C (x) 1.0 0.8 0.6 0.4 0.2 The function F C (x) used in the theory of Inverse Compton emission 0.0 0.0 0.2 0.4 0.6 0.8 1.0 A plot of the function F C x is shown at the left. This shows that there is a broad distribution of electron energies resulting from Compton scattering. x ( ε 1 /4 γ 2 ε) High Energy Astrophysics: Compton Scattering II 39/94

Rate of energy of scattered photons The above expression is more informative if we compute the rate of energy scattered per unit final photon energy per unit incident energy. This is 1 dn 3 ------------------- dtd 1 d 1 -- c T 4 ------------- n ---------- F C x 2 3c T nxf C x (54) High Energy Astrophysics: Compton Scattering II 40/94

x F C (x 0.20 0.15 0.10 0.05 0.00 0.0 0.2 0.4 0.6 0.8 1.0 The xf C x function is plotted at the left. This function gives a better idea of the rate of photon energy scattered to high energies as a result of Compton scattering. x( ε 1 /4 γ 2 ε) High Energy Astrophysics: Compton Scattering II 41/94

As you can see this function indicates a peak rate of scattered energy at 1 0.6 4 2 2.4 2 (55) Check on calculation As a check on this expression, one can evaluate the total energy radiated against the previous expression by integrating with respect to. 1 High Energy Astrophysics: Compton Scattering II 42/94

Since x 1 4 2, then 1 dn ------------------- d dtd 1 d 1 12 2 c T n 1 0 xf C xdx (56) The value of the integral is 1 9 Energy emitted per unit soft photon energy and 4 -- c 3 T The total power emitted comes from integrating over P 4 -- c 3 T 2 nd n 2 (57) and is (58) High Energy Astrophysics: Compton Scattering II 43/94

in agreement with the ultrarelativistic limit derived earlier. 3 Inverse Compton spectrum from a distribution of electrons 3.1 Direction of emission The goal of the following is to determine the emissivity of a distribution of electrons that are emitting photons via the inverse Compton process. This calculation is simplified by the fact that, in the lab frame, the emitted photons are scattered in the direction of motion of the electron. High Energy Astrophysics: Compton Scattering II 44/94

To see this consider a photon that is scattered at an angle cos 1 1 in the rest frame. In the lab frame 1 cos 1 1 + -------------------- 1 + 1 (59) When 1, for most angles 1, 1 1 (60) The only exception is when 1 1, then 1 1 (61) High Energy Astrophysics: Compton Scattering II 45/94

Since Thomson scattering in the rest frame is quasi-isotropic, most photons will be scattered in the forward direction given by 1 0. 1 This plot effectively illustrates this point. The consequence of this is that most of the photons are emitted in the direction of motion of the electron. This is similar to synchrotron emission. μ(lab frame) 1.0 0.5 0.0-0.5 γ 10 γ 2 Relationship between μ cos θ in lab and electron rest frames for relativistic scattering -1.0-1.0-0.5 0.0 0.5 1.0 μ(electron rest frame) High Energy Astrophysics: Compton Scattering II 46/94

B d As before let N be the number of photons per unit Lorentz factor. Then the number of electrons with Lorentz factors in the range d pointing in the direction of the observer is: N ---------- dd 4 (62) High Energy Astrophysics: Compton Scattering II 47/94

Hence the rate of energy per unit time per unit scattered photon energy radiated into this solid angle, produced by soft photons in the range d is 1 dn dtdd ------------------- N ---------- ddd 1 4 1 d (Note that we have multiplied by 1 associated with the number of photons.) (63) to determine the energy High Energy Astrophysics: Compton Scattering II 48/94

Hence, the emissivity of inverse Compton scattered radiation is 1 dn j 1 ----- 4 1 ------------------- Nd dtdd 1 3c T 1 ----------------------- 16 d n ---------- F C x N ---------- d 2 d (64) The emissivity therefore involves a double integration over the particle distribution and the input photon distribution. High Energy Astrophysics: Compton Scattering II 49/94

Integration over Lorentz factor We can do this part first since we generally have a good idea of what the particle distribution is. We assume, as before, that the particle distribution is given by N K e a min max (65) Remember that the energy of the incident and scattered photons satisfy: 1 ---- 4 2 (66) High Energy Astrophysics: Compton Scattering II 50/94

and that the mean energy of scattered photons is 1 4 -- 3 2 (67) so that if we are considering values of that approach either or then there will be end effects associated with min max the integral. We assume that this is not the case and estimate the inner integral F C x N ---------- d 2 (68) High Energy Astrophysics: Compton Scattering II 51/94

We calculate the integral by changing the integration variable to x 1 4 2. (69) Then 1 ----- 12 / 4 x 12 / d 2 N K 1 ----- 4 1 -- 1 ----- 12 24 a + 2 ---------------- 2 x a + 2 ----------- 2 / x 32 / dx (70) High Energy Astrophysics: Compton Scattering II 52/94

The initial limits on x are defined by min and max implying that 1 1 ----------------- x ---------------- 42 max 42 min (71) For the upper limit however, we also have the requirement x 1, so that x 1 4 -------------- 1 2 min (72) High Energy Astrophysics: Compton Scattering II 53/94

1 If ---- 4, as is usually the case, (that is we are considering very high energies 1 min ) then we have x 1 as the upper limit. 2 Making the above substitutions and using the derived limits: max min F c x---------- N d 2 K e ----- 1 ----- 2 4 a + 1 ------------------- 2 1 0 x a 1 ---------------- 2 FC x 2a K 2 + 4a + 11 e ------------------------------------------------------ 1 a + 1a + 3 2 ----- a + 5 4 dx (73) a + 1 ------------------- 2 High Energy Astrophysics: Compton Scattering II 54/94

Then, j 1 3c T 1 ----------------------- 16 n ---------- F C x---------- N d 2 d 3 --2 a 2 a 2 + 4a + 11 ------------------------------------------------------ a + 1a + 3 2 c a + 5 T max min a 1 ---------------- 2 ---------------- 2 (74) K e 1 a 1 nd W m 3 J 1 High Energy Astrophysics: Compton Scattering II 55/94

We write this more compactly as: j 1 fa a 1 fa c T K e 2 1 ---------------- max min a 1 3 --2 a 2 a 2 + 4a + 11 ------------------------------------------------------ a + 1a + 3 2 a + 5 ---------------- 2 nd (75) High Energy Astrophysics: Compton Scattering II 56/94

3.2 Emissivity in terms of photon number High energy observers count photons and photon-counting statistics come into the estimation of errors. Hence, the results of X-ray or -ray observations are often represented in terms of a photon flux: number of photons per second per unit area. Accordingly, the emissivity can also be represented in terms of number. Let dn -------------------------- dtdvdd Rate of emission of photons per unit time (76) per unit volume per unit energy per Steradian High Energy Astrophysics: Compton Scattering II 57/94

Then dn -------------------------- 1 dtdvdd 1 j 1 a + 1 fa c T K e 2 1 ---------------- max min a 1 (77) ---------------- 2 nd An important feature is a difference of 1 between the photon index and the spectral index. High Energy Astrophysics: Compton Scattering II 58/94

4 Features This expression for the emissivity has some important features: The expression for the emissivity is proportional to which is proportional to the Thomson cross section n e T L, where n e K e T L is the electron density. The Thomson cross section always figures in applications involving electron scattering. The emissivity is a power-law with exactly the same spectral index as the synchrotron emission. This is a tell-tale signature of inverse Compton emission. High Energy Astrophysics: Compton Scattering II 59/94

The dependence on the soft photon spectrum enters through the factor nd. The energy dependence of the inverse Compton emission is independent of the type of input photon spectrum. 5 Applications max min a 1 2 There are 2 traditional main applications for this formula. When the incident photon spectrum is a black body, e.g. the microwave background or approximating the emission from starlight by a black body When the synchrotron emission from a source produces a High Energy Astrophysics: Compton Scattering II 60/94

quasi-isotropic spectrum in the emitting volume 5.1 Evaluation of the soft photon integral We require where 0 a 1 2 nd (78) nis the number density of photons per unit energy. The energy density of photons per unit energy is u n n 1 u (79) High Energy Astrophysics: Compton Scattering II 61/94

We had previously defined the energy density per unit frequency by For the isotropic distribution we have used here, Moreover u d 1 u -- I c d u 4 ud u 4 ----- I c u d u ----- h d 1 (80) (81) (82) High Energy Astrophysics: Compton Scattering II 62/94

Therefore n 1 u 4 -------- 1 h 2 c I (83) 5.2 Blackbody distribution When the soft photon field is an isotropic black body distribution such as the microwave background, then the specific intensity is just given by the black body value: 2h I B 3 ------------ e c 2 h ----- kt 1 1 (84) High Energy Astrophysics: Compton Scattering II 63/94

The energy density per unit energy is then: Hence, 0 n a 1 2 8 ----------- h 3 c 3 kt a + 5 8 -------- 2 hc 3 e nd h ----- kt 1 2 x a + 3 2 0 1 x e x 1 1 d (85) (86) High Energy Astrophysics: Compton Scattering II 64/94

The integral Hence, j 1 x a + 3 2 e x 1 1 0 dx a ----------- + 5 2 a + 5 ----------- 2 fa 8 T ------------- h 3 c 2 a + 5 a + 5 ----------- a + 5 ----------- kt 2 2 2 ---------------- K e 1 (87) a 1 ---------------- 2 High Energy Astrophysics: Compton Scattering II 65/94

Group the numerical and a-dependent terms into, f bb a 3.2 a + 1 a2 + 4a + 11 ------------------------------------------------------- a + 1a + 3 2 a ----------- + 5 a + 5 ----------- a + 5 2 2 (88) The emissivity from a black body soft photon field and a relativistic electron distribution is therefore: j 1 T -----------f h 3 c 2 bb akt a + 5 a 1 ----------- ---------------- 2 Ke 2 1 (89) High Energy Astrophysics: Compton Scattering II 66/94

f bb (a) 25 20 15 10 5 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 a The function f bb a is plotted at the left. This can be used in estimates of the inverse Compton emission produced by the microwave background, for example. (90) High Energy Astrophysics: Compton Scattering II 67/94

5.3 Synchrotron Self-Compton emission In this case, the soft photons are synchrotron photons emitted from the same volume by the same electrons. Therefore, the power-law index of the soft photon input is a 1 2 and we can take u a 1 2. We suppose that the power-law holds between soft photon frequencies of and (l for lower and u for upper). Thus, n 1 u 1 0 u 0 ---- a + 1 0 l u 2 (91) High Energy Astrophysics: Compton Scattering II 68/94

where 0 is some fiducial soft photon energy. The integral, u a 1 2 n d a 1 0 l 2 u 0 u l ---- 1 d ---- 0 a 1 2 0 u 0 ln u l 0 (92) Notes: Because of the assumed power-law, the parameter a 1 2 0 u 0 is independent of 0. The parameter ln u l is known as the Compton loga- High Energy Astrophysics: Compton Scattering II 69/94

rithm. The upper energy can be determined from observation. The lower frequency may not be as well determined, although synchrotron self-absorption (to be treated later) sets a lower limit on. Fortunately, the evaluation of the l emissivity is not sensitive to either the upper or lower cutoff since they appear in a logarithmic term. High Energy Astrophysics: Compton Scattering II 70/94

Final form in terms of energy In the case of SSC emission, then, the emissivity is: j 1 a 1 fa c T K e 2 1 ---------------- max min a 1 ---------------- 2 nd fa c T K e 1 0 a 1 2u 0 u ln---- l (93) High Energy Astrophysics: Compton Scattering II 71/94

When we compare this with the expression for synchrotron emission from, say the isotropic version resulting from a tangled field j syn C 2 a------- Ke a + 1 0 c 0 e 2 2 a 1 2 (94) then we can see that the ratio of SSC to synchrotron emissivity can give us an estimate of the magnetic field since the factor of occurs in both expressions and therefore cancels out. K e High Energy Astrophysics: Compton Scattering II 72/94

Emissivity in terms of frequency Note that, for the inverse Compton emissivity, and Planck s constant combines with the factor as follows: j 1 hj 1 hu 0 u 0 u 0 (95) (96) High Energy Astrophysics: Compton Scattering II 73/94

Hence, a useful expression for SSC emissivity, when comparing different frequencies is jssc 1 fa c T K e 1 0 a 1 2 u u 0 ln---- l (97) High Energy Astrophysics: Compton Scattering II 74/94

5.4 Using synchrotron and self-compton fluxes to estimate the magnetic field The flux density from a stationary source at a luminosity distance,, is: D L 1 F ------- j dv D L 2 V (98) High Energy Astrophysics: Compton Scattering II 75/94

We assume a spherical source with constant electron density and uniform synchrotron emissivity. The flux density from the source at frequency is then 1 1 F 1 ------- fa c T K e 1 0 a 1 2ln u l D L 2 V u 0 dv (99) High Energy Astrophysics: Compton Scattering II 76/94

We could simply assume at this point that the photon energy density is constant throughout the volume. However, one of the exercises shows that: where u r u v 0 1 2 -- R r -- R + 1 -- 4 1 is the radius of the volume. 1 + ln ----------- 1 (100) High Energy Astrophysics: Compton Scattering II 77/94

Hence, u d V 4u 0 V 1 0 4u 0R 3 1 -- 2 R 3 u 0 0 R 0 1 -- 2 1 -- 4 1 1 + + ln ----------- 1 r 2 dr 1 -- 4 1 1 + + ln ----------- 1 2 d 3 --u 4 0 0V (101) High Energy Astrophysics: Compton Scattering II 78/94

The SSC flux, therefore is: SSC 3 -- 1 4D2 L F 1 ------- fa c T K e 1 0 a 1 ln u l u 0 V By comparison, the synchrotron flux is: F syn C 2 a e -------------- 2 D 2 ------- a + 1 Ke 0 c 2 0 2 a 1 ----------- ---------------- 2 V (102) (103) High Energy Astrophysics: Compton Scattering II 79/94

6 Models of synchrotron and SSC emission As above let us consider a spherical model in which the radius of the sphere is R. In the expression for the SSC flux SSC 3 F 1 -- 1 1 ------f 4D 2 ac T K e ----- 0 a 1 ---------------- 2 (104) ln u l u 0 0V we have the factor u 0 2 ----- Ipeak c (105) High Energy Astrophysics: Compton Scattering II 80/94

and I peak 2j syn R (106) Therefore, a model of the combined synchrotron and SSC spectrum involves the synchrotron parameters as well. When constructing models for the entire spectrum, we take this into account. An example is the following spectrum of Markarian 501. In this case, we have also had to allow for the fact that the Klein Nishina cross-section had to be used and the motion of the emitting region is relativistic. Also, the end points of the distribution in have to be included so that this example is quite a comprehensive one. High Energy Astrophysics: Compton Scattering II 81/94

MilliARC SEC 5 0-5 -10-15 -20 CONT: MKN501 IPOL 1634.000 MHZ MRK501,1.4+1.OGEOM.1 30 25 20 15 10 5 0 MilliARC SEC Center at RA 16 53 52.21671 DEC 39 45 36.6091 Cont peak flux 4.2070E-01 JY/BEAM Levs 1.000E-03 * (-1, 1, 1.500, 3, 5, 10, 20, 30, 50, 100, 200, 300) VLBI Image of Markarian 501 The X-ray and -ray emission originate from a region about an order of magnitude smaller than the smallest resolvable scale in this image. High Energy Astrophysics: Compton Scattering II 82/94

ε 2 dn/(dt d εda) (ev cm -2 s -1 ) 1e+03 1e+02 1e+01 B0.1 Gauss K e 4.3x10 5 cm -3 γ 2 3x10 6 R 5.3x10 15 cm 1e+00 1e+03 1e+05 1e+07 1e+09 1e+11 1e+13 ε(ev) X and TeV -ray spectrum of Markarian 501 High Energy Astrophysics: Compton Scattering II 83/94

7 Inferences from SSC emission on resolvable scales In general, we model synchrotron plus SSC spectra as we showed for MKN 501. However, when the emission is resolved it is possible to use both sets of data to estimate the magnetic field. Using the expressions for the synchrotron and SSC fluxes: High Energy Astrophysics: Compton Scattering II 84/94

F syn FSSC 1 C 2 a e -------------- 2 D 2 ------- a + 1 Ke 0 c 2 0 0.75fa 1 ------------------- D 2 c T K e ----- a 1 ----------- ---------------- 2 0 a 1 ---------------- 2 V (107) ln u l u 0 0V High Energy Astrophysics: Compton Scattering II 85/94

On dividing one equation by the other, a number of factors cancel ( K e D 2 3 V ) and -----------------. This gives: 0 c 2 -- m e ------ 2 r T 0 e 2 a + 1 ----------- 2 fa 0 ----------------- 2C 2 a r 0 ------ 1 ----- m e 0 a 1 ---------------- 2 a 1 F syn ------------- FSSC 1 ----------------- 2 u ln---- u 0 0 l (108) and, of course, B m e e 0. High Energy Astrophysics: Compton Scattering II 86/94

All of the parameters in the right hand side of the above equation are observable, except for. We shall see later how we can put constraints on. The parameter l u 0 (109) is also directly estimable from the peak surface brightness of the emitting region. l 2 ----- Ipeak c High Energy Astrophysics: Compton Scattering II 87/94

8 Example on Cygnus A hot spots X-ray image (contours) of Cygnus A overlaid on the radio image (greyscale) (Wilson, Young and Shopbell, 2000). The bright X-ray regions are coincident with the radio hot spots. High Energy Astrophysics: Compton Scattering II 88/94

Spectral break The radio and X-ray spectrum (in the form of fluence) of the brightest hot spot in Cygnus A. The bowtie represents the region of allowable X- ray slopes. High Energy Astrophysics: Compton Scattering II 89/94

The solid lines show model fits to the data. However, we can estimate the magnetic field using the data from this plot. For example take the point at 10 8.3 Hz Fsyn 10 10.8 10 26 W m 2 1 u 10 10 Hz 10 17.5 Hz 1 F 1 10 9.9 W m 2 0.55 (110) High Energy Astrophysics: Compton Scattering II 90/94

The following information on the radio source gives us the value of the energy density of the radio emission in the hot spot. This is obtained from the paper The Structure and Polarisation of Cygnus A at 3.6 cm, R.A. Perley and C.L. Carilli, in Cygnus A Study of a Radio Galaxy, ed. C.L. Carilli & D.E. Harris. Peak flux per beam 90 mjy Beam size 0.35 0.35 9 0 510 Hz (111) High Energy Astrophysics: Compton Scattering II 91/94

The value of the magnetic field, evaluated using equation (105) is 4 B 1.410 Gauss (112) Note that the slope of the spectrum implied by the bowtie is not the same as the slope of the radio spectrum. In the model shown in the figure, this is the result of the X-ray emission being affected by the end effects referred to earlier. Nevertheless estimates such as these are quite useful for the starting points of detailed models. High Energy Astrophysics: Compton Scattering II 92/94

By adopting an angular size for the hot spot we can estimate the minimum energy magnetic field. From the image, one obtains D hot spot 3.5 (113) The peak surface brightness of the hot spot can be evaluated from the flux per beam and the beam size [see equation (108)] and these parameters can be used to evaluate the minimum energy magnetic field. One obtains for the minimum energy magnetic field in the hot spot B min 9.210 Gauss (114) High Energy Astrophysics: Compton Scattering II 93/94 5

This is within a factor of 2 of the inverse Compton value. High Energy Astrophysics: Compton Scattering II 94/94