' ' # Thompson Scattering Consider a beam of radiation moving in the direction, and being scattered by an electron through an angle in the plane. The electron experiences the following electric fields which give rise to accelerations: # #$! " # #%& " The incident energy per unit area on the electron is given by the Poynting vector, ' (*) + %,.- # ()/ & + #, - Using Larmor s formula, and averaging over time gives the intensity of radiation scattered into the solid angle 021 043 05 043 05 021 # 021 67$89 + :<;>= + +" ) + ( 6@? 6 :<;>= + +" ) + ( 6@? It is clear that the scattered radiation is therefore :BACAD polarised even for unpolarised incident radiation, rising to polarised E AF at. Also note that as much radiation is scattered backwards as forwards. 021 021
G G K f w x g c g ] e h P x h X ] h ] h p ] G Considering unpolarised incident radiation (HJI K HML K HONQP ) define differential cross-section R2SUTWVXZY by e4f g fkj pqfre R2S[T K ]^U]_a`Cb fre _dc ]i_ g fqj ]_mlon c>^s` R2\ ^ut ]^ ]^U]_a`Cb yƒ ]_ ]_vcq_a]bc ybzq{o } ~i x t nl Integrating over X gives the total cross-section for scattering S[T K z>{@} w ~ x Thompson scattering assumes that } ~, so the recoil of the electron is negligable and the photon frequency is unchanged.
ž Compton and Inverse-Compton Scattering In Compton scattering high energy photons scatter off stationary electrons. They transfer energy and momentum to these electrons and so their wavelength increases. It is often succifient to transform into the zero-momentum frame, consider the collision as Thompson scattering and then transform back into the lab frame. This gives Š Š Œ Ž $ šu Zœ However, if in the zero-momentum frame Œ Ž d then the relativistic cross-section must be used, as given by the Klein-Nishina formula: Ÿ@ where Œ ƒ Ž / Z œ k Z œ The process of inverse Compton scattering considers collisions between high speed electrons moving at speed ª and low energy photons. Again it is often possible to transfer into the zero-momentum frame and consider the collision as Thompson Œ scattering (if «Ž ). Energy is lost by the electrons to the photons during this collision 4 5 ž4² &³µ $ ª $ «Z œ
¹ ¹ ¹ ¹ Ê The Cosmic Microwave Background After the Big Bang the Universe expanded and cooled. Until the Universe was 300,000 years old the temperature was so high that all baryonic matter was ionised. Therefore the mean free path for a photon was very small, and so the optical depth º!» ¼ ½. ¹ The radiation and matter were is almost perfect thermal equilibrium ¾ The radiation had a blackbody spectrum. At at redshift À ÁBÂ%ÂCÂ the electrons and ions in the plasma combined to form atoms ¾ the optical depth dropped dramatically º!» Ã Ã Á and their mean free path became much greater than the size of the Universe. ¹ The photons have continued to cool and are now at a temperature of Ä2ÅÇÆ (microwaves). Observing this CMB radiation allows us to directly observe the Universe at the epoch of recombination. ¹ Anisotropies in the CMB are present at the level of ÉZÊ À ÁBÂ2ËÍÌ. Detecting these anisotropies allows calculation of the geometry of the Universe, determination of mass densities, etc. Other radiation sources also lie along the line of sight and contaminate our measurements.
Î Î Î Î Î Foregrounds Galactic There are many sources of emission from the galaxy. This emission is generally on large scales similar to the interesting scales of CMB anisotropies. Remove contamination by observing away from the galactic plane and observing at many frequencies. Can then use spectral discrimination, ÏZÐOÑ Ò Ó Ô Õ (Ö Ø ÙuÚMÛ ). Dust The spectral index of optically thin dust is ÏvÜÞÝQßáà Ó Ô âãõ ä åçæ Free-free Optically thin so ÏZèè é ê4ëìê. Synchrotron Optically thin, fairly young sources so Ïvßáí$îðïñé ê4ëóòqô. Extragalactic sources Extra galactic sources may be optically thick or thin, free-free or synchrotron sources õ A wide variety of different spectra, so it is not possible to remove contamination spectrally. Î However, extragalactic sources generally have small angular sizes, so can remove them with high resolution imaging.
ø ø ø The Sunyaev Zel dovich effect Surface of last scattering ö ö ö öö öö öö z=1000 Plasma Cloud Clusters of galaxies have a hot (ùoú û übý%þ K) atmosphere of ionised gas. Densities are of the order of ü<ý%ÿ The high energy electrons will inverse-compton scatter CMB photons passing through the gas. ø The scattering optical depth is given by ú ú Even for a very rich cluster ú ü, so ú can be considered as the fraction of photons that undergo a scattering.
" On average the photons will be scattered to higher energies! The spectrum is shifted from a blackbody to higher energy, conserving photon number. At low frequency observe a decrease in intensity towards cluster; at high frequency observe an increase in intensity. The frequency where there is no intensity change is approximately 220 GHZ. Electrons scattered to higher energy Intensity SZ dip at radio frequencies Frequency
G D F C G G G # We can describe this intensity change as a change in brightness temperature, $ %, which in the Rayleigh-Jeans region -.0/21 $ % * +, 4 % & ')( 3 46587 9 % 48:; * +,=< where < is the Comptonization parameter. Find $ % >? mk for rich clusters. # Outside the Rayleigh-Jeans region $ @BA * F CED +?=H 7 C I!JLKNM * TVU -. % & CPO, H + Q H <R A % &SH # NB The above expression is independent of redshift. This means that the S Z effect is a very important tool for investigating early structure formation. # All the above neglects relativistic effects (approximately a,xw effect for a cluster at 10 kev.) # As well as this thermal S Z effect there is also a kinematic S Z effect due to cluster velocity along the line of sight. In the Rayleigh-Jeans region $ %VYZ\[^]`_à bczed $ % bcf^]hgi_akj * l2mnlloxp qsrxt O? lllul vxw x!y{z G -. % 4 O? l v~}^ H y{z
700.0 600.0 Flux Density at 900 λ (µjy) 500.0 400.0 300.0 0 1 10 Redshift Figure 1: Simulated S Z flux observed by the RT 900 baseline for a cluster of constant physical parameters projected back in redshift. ( ƒu ).
Combining with X-rays The hot gas also emits Bremsstrahlung radiation. Š ŒŽ u emission in the X-ray band. Spectral observations of knee in spectrum allows estimate of {Š. 8 gas is optically thin to X-rays Flat X-ray spectrum. Compare S Z and X-ray signals š œ {Š6 ŠžŸ Š N Š žÿ Assuming that Ÿh Ÿ (ie that cluster is spherical) and a geometry for the Universe (ª «) can calculate a value for Š P In theory can determine the geometry of the Universe from clusters at a range of different redshifts. Average over many randomly orientated clusters to reduce error due to aspherical clusters.
Source Subtraction Background sources will mask the S Z effect and their contamination must be removed. These confusing sources are generally much smaller than the size of a cluster, so can determine their flux by high resolution mapping. With an interferometer (eg Ryle) get this high resolution imaging from the long baselines which are insensitive to the large angular scale S Z effect. Subtract these sources from the short baseline data which is sensitive to the S Z. S Z effect Point Source Figure 2: Simulated flux observed by the RT against baseline for both a spatially extended S Z effect and for a point-like confusing source.
Å À À Glossary Absorbtion Coefficient, ² ³ : Fraction of incident radiation absorbed per unit length of absorber. Includes the effect of stimulated emission. Blackbody Radiation: Radiation produced when the emitters and the radiation are in prefect thermodynamic equilibrium. Intensity predicted by Planck s law. Brightness Temperature, µ ³ : The temperature that a blackbody would have to emit radiation of the observed intensity at a given frequency. Compton Scattering: The collision of a high energy photon with an electron. The photon looses energy and momentum to the electron and so its wavelength increases. Einstein Coefficients,! œ¹6µ 8 º¹»µ ḩ : Emission or absorbtion coefficients for a single emitter. Emissivity, ¼6³ : The emitted power per unit volume per unit frequency. Extinction: Attenuation (of starlight) due to absorbtion and scattering by the Earth s atmosphere and by interstellar dust. Flux Density: The power in radiation incident per area unit area per unit frequency. Often measured in Jy (½ ¾º ÁÀ ÂÄÃ!Æ Ç ).
Ó ð ä Í Ý Ý ñ ò Gaunt Factor: A quantum mechanical correction factor applied to formula for Bremsstrahulung radiation. Often É Ê. Inverse-Compton Scattering: The collision of a photon and a high energy electron. The electron looses energy to the photon whose frequency therefore increases. Kirchoff s Law: For a material in thernal equilibrium the absorbtion and spontaneous emission coefficients are linked by ËœÌ Í ÎšÌLÏ ÌÐ Ñ Ò Larmor s Formula: Equation relating the total radiation rate and the acceleration of a charged particle ÔSÕ8Ö Õ6 ÙØŽÚ6Û Õ ßáàuâiãåäçæ Ü ÝŽÞ Ú Ý Luminosity: Power emitted in radiation. Mass Absorption Coefficient, è Ì : The total cross-section per unit mass of material è Ì Í Î Ìêéœë. Optical Depth, ì Ì : A measure of the integrated opacity along a path through a layer of material at a particular frequency ì Ì Í í ΚÌ~îï. Planck Function: Formula that determines the distribution of intensity of radiation under conditions of thermal equilibrium at temperature Ñ. Ï Ì Ð Ñ Ò Í ð»ñ Ì æ ó ô8õ~ö ̺ 6øùûúüþý ò Rayleigh-Jeans Approximation: Low frequency approximation to the Planck formula, appropriate when ÿ Ñ. ø»ú. Ï ÌÐ Ñ Ò É
Source Function, : The ratio of emissivity to the opacity of a material. For the case of thermodynamic equilibrium. Specific Intensity, : The flux density per unit solid angle. Spectral Index, : Spontaneous Emission Coefficient, : Power emitted spontaneously per unit volume per unit frequency per unit solid angle. Stoke s Parameters: Four parameters (I, Q, U, V) which characterise a beam of polarised light. Surface Brightness: Emergent specific intensity. Synchrotron-Self-Compton: Radiation emitted through the synchrotron process which is then inverse-compton scattered to higher frequency by the same population of electrons responsible for the original emission. Thermal Radiation: Radiation from emitters in thermal equilibrium. If the radition and emitters are also both in equilibrium then the radiation is blackbody.