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Radio astronomy 29:186 Spring 2012 Homework 2 key: Radiation processes, Larmor formula Define useful constants, conversions (N.B. not all are needed for any given assignment) 23 J kB 1.38 10 5 σsb 5.67 10 K 34 h 6.6260693 10 26 cm s K sr W 26 RE 6378 km 3 mJy 10 2 Jy m Hz 31 27 kg 19 J mp 1.67 10 3 amp 19 kg qe 1.602 10 6 keV 10 eV 7 newton 2 μ0 4 π 10 8 AU 1.5 10 km 4 pc 3.06 10 m Jy 10 me 9.109 10 eV 1.6 10 2 16 joule sec Lsun 4 10 W erg MeV 10 eV coul 5 Rsun 7 10 km 12 farad ε0 8.85 10 m 1.[1] Estimate the number density of cosmic microwave background (CMB) photons, assuming (correctly) that the CMB radiation originated from a nearly perfect black body emitter at a temperature of 2.72 K. Express in cm-3. Hint: Find the energy density, and assume each photon has exactly kT energy. T 2.72 K ucmb 4 4 σsb T c Ecmb kB T ra-spr2014-hw2-key.xmcd 13 erg 3 ucmb 4.141 10 ncmb ucmb Ecmb 1 cm 3 3 ncmb 1.103 10 cm 2/21/2014 2. [1] The Green Bank telescope (diameter = 100m) switches between a radio source unknown flux density and a resistive load with a physical temperature T = 250K. The power received by the radio source and the resistor are equal. What is the flux density of the radio source (in Jansky)? Assume the telescope collects 70% of the incident flux of the source. T 250 K Pres 3.45 10 Hz π 2 Aeff 0.7 d 4 d 100 m S 21 W Pres kB T Pres Aeff S 62.753 Jy 3. [1] A ball with a total charge of 1 coulomb and a mass 1 kg is dropped from the top of a tall building of height 100 m. What is the total power radiated as a function of height? (Ignor air friction). Q 1 coul a 9.8 m 2 s 2 2 2 Q a 1 P 3 c3 4 π ε0 P 2.137 10 14 W Power is constant until it hits ground! 4. [2] This problem calculates the power radiated from thermal electrons in the lower solar corona. The plasma has a mean temperature 105 K, and the electrons are immersed in a magnetic field of field strength B = 1 Gauss. (a) What is the mean velocity of the electrons? 5 T 10 K E 3 k T 2 B V 2 E me 3 km V 2.132 10 sec (b) What is the mean acceleration of each electron? B 1 gauss F qe V B ra-spr2014-hw2-key.xmcd a F me 13 m 2 a 3.749 10 s 2 2/21/2014 (c) What is the total power radiated per electron? 2 2 2 qe a 1 Pelec 3 c3 4 π ε0 27 Pelec 8.027 10 W (d) Assume a mean density 1012 elec/cm3 and a thickness 0.1Rsun starting at the base at the solar photosphere. What is the total radiated power? Compare with the observed solar x-ray luminosirty (Lx = 1027 ergs/sec) 5 Rsun 7 10 km 12 3 ne 10 cm ΔR 0.1 Rsun 27 erg Pxray 10 s 2 Volume 4 π Rsun ΔR Ne Volume ne 44 Ne 4.31 10 25 erg Ptotal Pelec Ne Ptotal 3.46 10 sec Ptotal Pxray 3.46 % 5.[2] A linear accelerator of length 10 m uniformly accelerates protons to kinetic energy 100 MeV. Ignore relativistic effects. 2 L 10 m E 10 MeV (a) What is the power radiated by each proton (Watts)? vfinal 2 E mp vfinal 0.462 c 2 a vfinal 2 L 14 m 2 a 9.581 10 s 2 2 2 qe a 1 P 3 c3 4 π ε0 P 5.241 10 24 W (b) What fraction of the energy imparted to the protons is lost to radiation? ra-spr2014-hw2-key.xmcd 3 2/21/2014 2 L 7 t 1.445 10 s a t Erad P t total energy radiated: Note that power radiated is constant along path since acceleration is constant 18 Erad 4.733 10 Erad E MeV 18 4.733 10 % Not much! (b) Plot the normalized (1= max) power pattern of the radiation [use a polar plot, indicate the direction of motion of the protons]. Tip:If using MathCAD, the poalr plot option will be useful P ( θ) sin ( θ) ra-spr2014-hw2-key.xmcd 2 θ 0 0.001 2 π 4 2/21/2014 120 90 1.2 60 1 0.8 150 30 0.6 0.4 0.2 P ( θ) 180 0 0 210 Proton velocity is along θ = 0 330 240 300 270 θ 6. [2] A white dwarf star with surface temperature 85,000K and radius = 1 Earth radius i surrounded by an ionized sphere shell of HII gas of radius 0.1 pc. Assuming the tempera the gas is 10,000K. (a) What fraction of the star's radiation is capable of ionizing neutral hydrogen? Plot the Planck function vs. frequency; indicate the region of ionizing photons with a vertical line (use the 'marker' option in MathCAD plots). 3 h ν ν B ( ν T) 2 exp h 1 2 c kB T νLy c 912 Angstrom ra-spr2014-hw2-key.xmcd 1 Twd 85000 K ν 0.1 νLy 0.11 νLy 10 νLy 5 2/21/2014 4 1.510 110 1 4 B ν T wd 510 5 0 0 2 4 6 8 ν νLy 100 νLy 1 νLy B ν Twd dν 84.573 % 100 νLy 0.01 νLy B ν Twd dν (b) What is the mean electron density in the ionized sphere? Assume the recombination coefficient is α = 3 x 10-14 cm-3 sec-1. Rs 0.5 pc B ( T) Rwd RE σsb 4 T π ELy 13.6 eV NLy 0.85Lwd ELy 2 3 ne 10 cm ra-spr2014-hw2-key.xmcd 14 αH 3 10 Lwd B ( T) 4 π Rwd 2 3 1 cm sec Lwd 2.306 Lsun RHII 0.1 pc 44 1 NLy 3.604 10 s initial guess value 6 2/21/2014 Given 3 NLy RHII = 4 π α n 2 H e 1 3 3 Find ne 316.373 cm ra-spr2014-hw2-key.xmcd 7 2/21/2014 4 π 3 27 ΔR 1.437 10 L 3 2 29 2 π Rsun ΔR 2.155 10 ra-spr2014-hw2-key.xmcd L 8 2/21/2014 is ature of ra-spr2014-hw2-key.xmcd 9 2/21/2014