13 Oct 2008 Ay 20 E.S. PhinneyBasic Astronomy and the GalaxyProblem Set 2 Due in class Monday 20 October, 2008Homework Problems:1. Angular resolutions of radio and other telescopesCalculate, in arcseconds, the angular resolutions of some famous diffraction-limited telescopes:a) Caltech’s 40 m single-dish radio telescope in Owens Valley, operating at the 1.4 GHzfrequency of the neutral hydrogen transition.b) Caltech’s CARMA array of mm-wave telescopes, with longest baseline of 2 km, operatingat the 115 GHz frequency of the ground-state transition of Carbon monoxide.c) The Arecibo single-dish radio telescope (1000-foot diameter) in Puerto Rico, operatingat 1.4 GHz frequency.d) The national Very Large Array of radio tele sc opes in New Mexico, with longest baselineof 36 km, operating at 1.4 GHz frequency.e) The national Very Long Baseline Array, whose radio telescopes c over the continentalUS, with additional stations in Hawaii and the Virgin Islands, operating at 1.4 GHzfrequency.f) The 2.4 m Hubble Space Telescope, observing at the shortest wavelength of its latelamented ACS camera, 380 nm.g) The 0.85 m Spitzer Space Telescope, operating at the shortest wavelength of its imagingphotometer, 24µm.h) The planned Caltech TMT (Thirty Meter Telescope), with the adaptive optics instru-ment NFIRAOS (high-speed deformable mirrors and six laser-created guide ‘stars’ in theupper atmos phere) cancelling earth’s seeing at the shortest well-correctable wavelength,1.0 microns.i) NASA’s planned James Webb Space Telescope at its shortest optimised wavelength,1.2 microns.2. What we’d like to see Calculate the angular sizes of1a) A sun-like star at 100pc distance from earth.b) A red-giant star with radius 1000 times the sun’s radius, at 100pc distance from earth.c) The maximum separation of a pair of sun-like stars orbiting each other with a 5-dayorbital perio d, at 100pc distance from earth.d) The maximum separation of a pair of sun-like stars orbiting each other with a 100-yearorbital perio d, at 100pc distance from earth.e) The shell of a supernova (exploding at 104km s−1), at 10 Mpc distance,one month afterthe explosion.11kpc = 103pc, 1Mpc =106pc, 1Gpc=109pc1f) The region of a galaxy at 100 Mpc distance around a 108Mblack hole where the starshave orbital velocities greater than 500km s−1(i.e. much greater than found elsewherein galaxies).g) A newly formed cluster of stars (diameter 2 pc) in a pair of me rging galaxies at 1 Gpcdistance.3. Fading into the distancea) The faintest objects the Palomar 60-inch telescope can detect in imaging with reasonable(several minute) exposures is a Johnson-Vega magnitude mV= 22. Out to what distancecould it detect (i) A star like the sun, (ii) a white dwarf with the same surface temperatureas the sun (white dwarf radii are 0.01 of the sun’s radius), (iii) a quasar with luminosityνLν= λLλ= 1046erg s−1at all optical and ultra-violet wavelengths?b) The heroic Hubble Ultra-Deep Field imaged a single patch of sky continuously for 400orbits, of which 56 (135,320 seconds) were in a ‘V’-like band (606nm). The faintestdetected sources in that ‘V’ band image had an AB magnitude (BEWARE! not JohnsonVega scale!) at 606 nm of about 27.5. At what distance (in pc) would such a heroicexposure be able to detect (i) A star like the sun, (ii) a white dwarf with the same surfacetemperature as the sun (white dwarf radii are 0.01 of the sun’s radius), (iii) a quasarwith luminosity νLν= λLλ= 1046erg s−1at all optical and ultra-violet wavelengths?4. Spectroscopy is a lot slower than imaginga) Compute the specific flux Fν(in Jy, and erg/cm2/s/Hz) and Fλin erg cm−2s−1˚A−1froma V = 22m galaxy. What is the rate at which V-band photons from this object in theV band strike the mirror of the Hale 200-inch-diamter telescope from this object?b) A spectrum of this galaxy is taken. The effective entrance aperture of the spectrographis 2 × 2arcsec2. The surface brightness of the night sky at Palomar in the V band ona dark night is 20.4 mag/arcsec2 (i.e., 1 arcsec2 emits a flux equivalent to that of a V= 20.4 mag object). What is the effective V magnitude of the foreground sky patch asseen by the spectrograph in its aperture?c) Now assume that the overall efficiency of the telescope + spectrograph + detector is10%. A resolution element in the spectrum is 10˚Awide, and we are interested in theregion around λ = 5500˚A. How many counts per resolution element are detected fromthe galaxy spectrum alone in a 1- hour exposure? From the foreground sky?d) If a blank piece of sky is measured at the same time in order to subtract the sky spectrumfrom the total, what is the signal-to-noise ratio per resolution element in the final, sky-subtracted galaxy spectrum? (Neglect the detector noise, and assume Poisson photonstatistics.)5. Zodiacal light and the search for extrasolar earth-like planetsDust released into the solar system from asteroid collisions and comets slowly spirals into thesun. It was first discovered because it scatters sunlight, creating a ‘glow’ in the dark nightsky.a) Astronomers quote the surface brightness of the z odiacal light in the anti-solar directionas seen from earth as 60 S10’s. S10is an awful unit, equal to the (V-band) surfacebrightness that one would get by spreading the light from a star of mV= 10 over 12square degree. Calculate the flux of zodiacal light from the dark night sky, and thuscompute the fraction of the sun’s light scattered by the zodiacal dust located beyondthe earth’s orbit.b) What fraction of the sun’s light is scattered by the earth?c) At a distance of 10 pc, what magnitude would be an earth-like planet in a 1AU orbitaround a sun-like star, when it is at its largest elongation from its star?d) The zodiacal light is observed to have about the same optical colors as sunlight, whichimplies that the dust grains must be larger than optical wavelengths, i.e. their sizea>∼1µm. What is an approximate lower limit to the ratio of the total mass of zodicaldust particles to the mass of the earth (assume that most of the zodical dust particlesare within 2 AU of earth)?e) Suppose we obse rve with a 10 m diameter space telescope, in green light, a sun-like star,10 pc distant, orbited by a solar system identical to ours. What is the ratio of the fluxfrom that solar system’s zodiacal light in the resolution element containing its earth, tothe actual flux from its earth, when its earth is at the