High Energy Astrophysics
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1 High Energ Astrophsis Essentials Giampaolo Pisano Jodrell Bank Centre for Astrophsis - Uniersit of Manhester giampaolo.pisano@manhester.a.uk - Februar 01
2 Essentials - Eletromagneti spetrum - Radiation propagation & definitions - Radiatie transfer - Thermal and Blak-Bod radiation - Speial relatiit formulae Referenes: - Longair - Vol 1, Chapter 1 - Rosswog & Bruggen - Par. 3.
3 Eletromagneti spetrum 1/ Radio mm/sub-mm FIR MIR NIR V NUV UV Soft X-ra Hard X-ra γ-ra λ[ m] If radiating matter is in thermodnamial equilibrium: [ Hz ] - Useful formulae: λ - Waelength Frequen relation [m s -1 ] : Speed of light in auum eV 16. C V 16. J - Energ gained b an eletron falling through a potential differene of 1 Volt E h E[ ev ] [ Hz] - h [J s] : Plank onstant - Energ assoiated to a photon of frequen E 5 ~ kbt E[ ev ] T[ K] - Relation between energ and assoiated temperature - k B [J/K] : Boltzmann onstant
4 Eletromagneti spetrum / Radio mm/sub-mm FIR MIR N I R V N U V UV Soft X-ra Hard X-ra γ-ra λ[ m] [ Hz ] Radio 3MHz 30GHz 0m λ 1m T ~ ( 4 1 K mm & submm 30 GHz 3 THz mm λ 0. 1 mm T ~ (1 0 K Infrared 3THz 300THz 0µ m λ 1µ m T ~ ( 4 K Optial Hz Hz 1µ m λ 300nm T ~ 4 4 K Ultraiolet 15 Hz 3 16 Hz 300nm λ nm T ~ ( 5 6 K X-ra Hz 3 Hz nm λ 0. 01nm T ~ ( 6 9 K γ-ra 19 3 Hz λ 0. 01nm T 9 K We will talk about obserations and related tehnologies in details later on
5 Essentials - Eletromagneti spetrum - Radiation propagation & definitions - Radiatie transfer - Thermal and Blak-Bod radiation - Speial relatiit formulae Referenes: - Rbiki & Lightman - Chapter 1 - Rosswog & Bruggen - Par. 3.3
6 Radiation propagation & definitions - 1/6 - For a plane wae, the amount of energ passing through element area da in time dt is: da (dt de nˆ de F da dt de Energ W F - Energ flu da dt Time Area m m - The energ passing through element area da in time dt in the frequen interal [,+d] is: da de nˆ de F da dt d ( dt, d F de da dt d Energ Time Area Bandwidth -Monohromati Energ flu W m Hz
7 Radiation propagation & definitions - /6 - The total energ flu is obtained integrating oer the frequenies: F 0 F d Energ Time Area - Total energ flu W m - For a spheriall smmetri isotropi soure: Conseration of energ: F( r π πr 14 r1 F( r 4 r r 1 F ( r 1 r1 onst F ( r - Inerse square law r r
8 Radiation propagation & definitions - 3/6 - Energ through area da, within solid angle dω from nˆ in time dt and frequen range d : da ( dt, d dω de nˆ de I da dt d dω I de da dt dω d Time Area Energ Solid angle Bandwidth m W Ster Hz - Speifi Intensit (Brightness with I I ( ϑ, ϕ - Integrating oer all solid angles: J 1 Energ I ( Ω dω 4π Time Area Bandwidth - Mean Intensit W m Hz
9 Radiation propagation & definitions - 4/6 - Gien a radiation field with angular dependene I (ϑ,ϕ, the ontribution to the energ flu in the diretion nˆ from the solid angle dω at ϑ is: da ( dt, d dω da nˆ ϑ df de I ( ϑ, ϕ daosϑ dt d dω (Note that daosϑ da is the projeted area - In terms of flu de da dt d we hae: df I ( ϑ, ϕ osϑ dω - Integrating oer all solid angles: F Energ ( nˆ I ( ϑ, ϕ osϑ dω Time Area Bandwidth - Net flu in diretion n^ W m Hz
10 Radiation propagation & definitions - 5/6 - Consider a linder about a ra in diretion with olume dvdadt ; the amount of energ rossing da in dt and in d is: nˆ dt da dω ( d nˆ de but de u ( Ω da dt dω d ( de I da dt dω d u I ( Ω ( Ω Energ Volume Solid angle Bandwidth - Integrating oer all solid angles: m 3 J Ster Hz - Energ densit per unit solid angle u 1 4π Energ I( Ω dω J Volume Bandwidth - Speifi energ densit J m 3 Hz
11 Radiation propagation & definitions - 6/6 Brightness onstan along ras in free spae - Consider two surfaes da 1 and da normal to a main ra; the energ passing through them is: r da d Ω1 r da da 1 da d Ω r 1 da da 1 de I da dt dω Ω1 d I da 1 da I 1 da da dt r d da1 de I da dt dω d I da dt d r 1 - Energ onseration: de 1 de I 1 I Brightness is onstant along a ra
12 Essentials - Eletromagneti spetrum - Radiation propagation & definitions - Radiatie transfer - Thermal and Blak-Bod radiation - Speial relatiit formulae Referenes: - Rbiki & Lightman - Chapters 1 & 4 - Rosswog & Bruggen - Par. 3.4
13 Radiatie transfer 1/4 Matter or plasma an inrease or derease the brightness of a Spontaneous emission radiation field b emission or absorption - The energ added b the isotropi emission of the olume dvda d is: d da ( dt, d dω de de ε dv d Ω dt d ε da d d Ω dt d ε : Emissiit - From the brightness definition: di de ε da d dω dt d ε d da dω dt d da dω dt d ε Energ Time Volume SolAng BW Brightness Length m di ε d - Intensit added to the beam in d 3 W Ster Hz
14 Radiatie transfer /4 Absorption - Imagine absorbing partiles with ross-setion σ and densit n : n d da dω ( dt, d de Total absorbing area datot σ n da d da Tot da Tot - B definition: - The energ absorbed from the beam is: I da dω dt d I de Tot ( n da d dω dt σ d di de I n da dω dt d da d dω dt da dω dt d σ d I nσ d α nσ Absorption oeffiient di α Id - Intensit loss of the beam in d [ ] [ ] 1 1 Length m
15 Radiatie transfer 3/4 - Combining the brightness emission and absorption: di di ε d α I d di d α I + ε - Radiatie transfer equation (0 I ( I Emission onl α 0 di d ε o ' Absorption onl ε 0 di αi d I + ( I (0 ε ( ' d' 0 I I (0 ep α ( ' d' 0 ( - Brightness inrease equals emission integrated along line of sight - Eponential dea with absorption integrated along line of sight
16 - Optial depth Radiatie transfer 4/4 o - Let s define: dτ α d or τ ( α ( ' d' - Optial depth 0 ' - A medium an be: τ τ >>1 <<1 - Optiall thik (opaque - Optiall thin (transparent Absorption onl - Substituting τ : I I ( I (0 ep( τ I (0 ep α ( ' d' 0 ( τ >>1 I ( 0 τ <<1 I I (0 - Aerage distane traelled b photon in absorbing media without being absorbed: ( l 1 1 α nσ I I - Mean free path
17 Radiatie Transfer in Astrophsis Interstellar Medium (ISM Dust in ISM - Infrared emission and absorption Star formation region - Starlight optial absorption - Emitting plasma - Emission/absorption of gases, dust or plasmas - From star interiors to the Big Bang
18 Essentials - Eletromagneti spetrum - Radiation propagation & definitions - Radiatie transfer - Thermal & Blak-Bod radiation - Speial relatiit formulae Referenes: - Rbiki & Lightman - Chapter 1 - Rosswog & Bruggen - Par Bradt - Chapter 6
19 Blak-bod radiation 1/4 - Thermal radiation is radiation emitted b matter in thermal equilibrium - Blak-bod radiation is radiation whih is in thermal equilibrium with matter - Blak-bod radiation an be obtained keeping radiation inside an enlosure at T until equilibrium is ahieed: T T I B (T (T B - Small hole to measure the radiation without disturbing equilibrium Blak-bod brightness - Independent enlosure properties/shape - Dependent onl on the temperature T - Homogeneous and isotropi I B (T
20 Blak-bod radiation /4 Plank spetrum - The brightness of a Blak Bod is: 3 h B ( T e 1 h kt - Plank law 1 (T B Raleigh Jeans Wien h kt e Low h << kt h kt e 1+ h kt I ( T R J kt - Raleigh-Jeans law High h >> kt h kt e 1 e h kt I 3 W h kt ( T e - Wien law Ma λ ma T 0.009[ m K] - Wien displaement law
21 Blak-bod radiation 3/4 - BB spetra at different T - Integrating, the Total Flu is: F 4 B osϑ d dω σt - Stefan-Boltzmann - σ : Stefan-Boltzmann onstant law - Monotoniit with T : eer B (T ure lies entirel aboe all the others at lower temperatures - For a gien I we an define a T b : - In the R-J limit: h << kt k T b I I B ( T b T b : Brightness Temperature
22 Blak Bod Radiation in Astrophsis 1/ The Sun - Emission peak in optial region - Blak Bod equialent temperature T5777 K
23 Blak Bod Radiation in Astrophsis / Cosmi Mirowae Bakground Radiation - Emission peak at mm waes: - Best known blak bod soure - T.736K - Flutuations are the seeds of present large sale strutures
24 Essentials - Eletromagneti spetrum - Radiation propagation & definitions - Radiatie transfer - Thermal & Blak-Bod radiation - Speial relatiit formulae Referenes: - Rbiki & Lightman - Chapter 4 - Rosswog & Bruggen - Chapter 1 - Bradt - Chapter 7
25 Speial relatiit 1/7 - Coordinates relations between frames - Consider rest frame S and frame S moing with relatie speed along diretion: - Spae-time oordinates transform as: Lorentz Transformations z z S ' S where ' γ ( ' + t' ' γ ( t ' ' z z' and z' z t γ ( t' + ' t' γ ( t γ Lorentz transformations 1 1 β - Lorentz fator and β - Length L in S measured in S with t0: L' ' 1 ' γ ( 1 γl L L' / γ - Length ontration - Time interal T in S with 0 measured in S: T ' ' t t1 γ ( t t1 γ T ' T γ T ' - Time dilation
26 Speial relatiit /7 Veloities Transformations - Veloities relations between frame S and S - Differentiating oordinates in S: d γ ( d' + dt' d d' dz dz' dt γ ( dt ' + d ' d γ ( d' + dt' u dt γ ( dt' + d' d d' u dt γ ( dt' + d' dz dz' u z dt γ ( dt' + d' ' S S ' d u, u... dt' u u u z u + 1+ u u γ (1 + u u z γ (1 + u -Veloities transformations - Similarl for oordinates in S : u u 1 u u u γ (1 u uz uz γ (1 u
27 Speial relatiit 3/7 - E and B relations between frame S and S Fields Transformations S S ' ' - The deriation gies: - Fields transformations + ( ( ' ' ' z z z B E E B E E E E γ γ + ( ( ' ' ' z z z E B B E B B B B γ γ
28 Speial relatiit 4/7 - Energ and momentum relations between frame S and S Energ & Momentum + + ' ( / ( p E E E p p γ γ - Energ & momentum transformations - These are: ( / ( p E E E p p γ γ - Relatiisti quantities and relations: m γm 0 - Mass 0 m m p γ - Momentum 0 m m E γ - Energ 0 1 ( m K γ - Kineti energ ( m 0 : rest mass 0 ( ( m p E +
29 Speial relatiit 5/7 Propagation of light - Doppler effet - Periodi pulses in S with: T π / ω S S Obserer in S - Assume two pulses emitted in 1 and : t T 1 - In S, for time dilation: t 1 γ t πγ / ω - Differene in obsered arrial times in S: t arr t1 ω π t arr d t (1 - Obsered frequen in S: 1 os 1 ϑ πγ (1 osϑ ω ω ω - Relatiisti γ ( 1 osϑ Doppler formula d l 1 l t d t Relatiisti Doppler as ombination of time dilation and light propagation times but 1 1 ϑ osϑ
30 Speial relatiit 6/7 Aberration of light - Photon emitted at a ertain angle in moing frame S appears under a lower angle in rest frame S: - Photons eloit omponents: u u u in S u u' + 1+ u' u γ (1 + u osφ sin Φ - Using the eloit transformations: osφ + 1+ osφ sin Φ γ (1 + osφ ' S S u tan Φ u r u Φ ' sin Φ γ (1 + osφ S u r Φ (1 + osφ osφ + sin Φ tan Φ - Aberration of γ (osφ + Light formula
31 Speial relatiit 7/7 Relatiisti beaming - Distortion of the beam of a fast moing radiating partile - Consider a relatiisti partile emitting isotropi radiation in S : S S Φ π/ Φ 1/γ γ >>1 - The aberration formula for relatiisti speed and Φ π beomes: Φ tan Φ sin Φ 1 γ (osφ + γ Φ 1 γ In the rest frame the radiation looks onentrated in the forward diretion with half of the photons ling within a one of semi-angle 1/γ
32 Essentials - Eletromagneti spetrum - Radiation propagation & definitions - Radiatie transfer - Thermal & Blak-Bod radiation - Speial relatiit formulae Radiation Proesses
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