Relativity and Astrophysics Lecture 10 Terry Herter. Doppler Shift The Expanding Universe Hubble s discovery

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1 Doppler Eet Doppler Eet Relatiity and Astrophysis Leture 0 Terry Herter Outline Doppler Shit The Expanding Unierse Hubble s disoery Reading Spaetime Physis: Chapter 4 Problem L-, page (due today/monday) Will hand bak on Monday i you hand it in on today Pim Wednesday, Sept. 3 Closed book and notes, will oer material through today Should know Spaetime interal, undamental postulates o Speial Relatiity, simultaneity and other issues raise by atiity Will hae both qualitatie and quantitatie questions Lorentz Transormation equations will be proided (i needed) A90-0 Doppler Eet

2 Doppler Eet Doppler Shit The Doppler eet is extremely important or Astronomy. This is a shit in the requeny (waelength) o light rom an atom due to motion with respet to the erer. It is the Doppler eet the allows measurement o the expansion o the Unierse (resulting rom the Big Bang) Following problem L-5 in the textbook we will derie the Doppler shit. A90-0 Doppler Eet 3 Doppler Shit 3 0 <= pulse number x = t-3/ t-/ t-/ t Consider a roket whih has soure that s pulse with a requeny (pulses or waes per seond). Suppose these waes trael in the positie x-diretion with a speed. Thus distane between pulses will be / (eloity time). Setting up the origins Let the 0 th pulse pass the origin (x = 0) at t = 0 and hoose the origin o the roket to pass the origin o the laboratory at the same time. Lab Frame: The 0 th pulse moes to the right with the speed o light, so its position at any gien time x = t. The next pulse is delayed by a distane/time /, while the nd pulse is delayed by /, et. Thus the position or the n th pulse is gie by x t n n t x For the lab rame. x A90-0 Doppler Eet 4

3 Doppler Eet Doppler Shit 3 0 <= pulse number x = t -3/ t -/ t -/ We an perorm the same argument to get the equialent ationship or the roket rame. x t n n t x Now apply the inerse Lorentz transormation (moing rom the roket rame to the lab rame). t t x x x t To get n t x x t Substituting or gies n t x t x t x A90-0 Doppler Eet 5 Doppler Shit Set the two equations or n equal (sine they muh orrespond to the same pulse (eent) gies: For pulse going in the negatie x-diretion We hae the equialent equations but the pulse is moing to the let. n t x n t x Using the Lorentz transormation and ollowing the same logi we arrie at the shit in requeny or pulses traeling to the let. These two equations are equialent. A sign aounts or whether the soure is moing towards you or not (by onention positie eloities are those that are moing away) A90-0 Doppler Eet 6 3

4 Doppler Eet Redshit Astronomers deine the redshit, z, o an objet as: Note that z z I the redshit is known we an sole or the eloity o the objet. This gies z z Quasars hae been measured to ery high redshits. For instane, a quasar measured to hae z = 4.897, the aboe equation gies a eloity (atie to the speed o light), = 0.944! A90-0 Doppler Eet 7 Redshit at low eloities At low eloities z 4 Ignoring terms greater than linear in, z Astronomers typially use the onentional eloity rather than one atie to the speed o light. This then beomes z on For nearby objets (galaxies) this approximation works well. A90-0 Doppler Eet 8 4

5 Doppler Eet Early Astronomial History Slipher (~9) notied that spiral nebulae showed almost predominantly redshits. By 95 he had radial eloities or 40 galaxies Hubble used the 00-inh telesope on Mt. Wilson to measure distanes to 8 galaxies Found linear ation between inreasing redshit and inreasing distane, now known as Hubble s law H o d on ~ z Hubble ound that the Unierse is expanding! The greater the distane, the higher the reession eloity A major goal o HST was to measure H o aurately. Currently, H o ~ 70 km/s/mp Oten used is H o = 00 h km/s/mp = (0 0 yrs) - h A90-0 Doppler Eet 9 5