Today s class. The optical Doppler effect Excursion to the universe!

Transcription

1 Today s class The optical Doppler effect Excursion to the universe!

2 The Doppler Effect We are all familiar with the acoustic Doppler effect. It relies on the speed of sound in air. For light: No ether! We expect some differences. (Also must include time dilation!)

3 How about the optical Doppler effect? Δt: proper time between source λ = c Δt Time between crests: Δt = γ Δt (time dilation) v λ =? 2 1 v Δt 1 was emitted from here 2 was emitted from here

4 How about the optical Doppler effect? Time between crests: Δt = γ Δt (time dilation) v λ =? 2 1 v Δt 1 was emitted from here 2 was emitted from here

5 How about the optical Doppler effect? Time between crests: Δt = γ Δt (time dilation) v λ =? 2 1 v Δt 1 was emitted from here 2 was emitted from here

6 How about the optical Doppler effect? Time between crests: Δt = γ Δt (time dilation) v λ =? was emitted from here v Δt c τ c (τ - Δt ) λ = c τ - c (τ - Δt )-v Δt = c Δt - v Δt = Δt (c - v) = γ Δt (c - v) 2 was emitted from here

7 How about the optical Doppler effect? λ = c τ - c (τ - Δt )-v Δt = c Δt - v Δt = Δt (c - v) = γ Δt (c - v)

8 How about the optical Doppler effect? λ = c τ - c (τ - Δt )-v Δt = c Δt - v Δt = Δt (c - v) = γ Δt (c - v) f obs The observed frequency (from the track) is: c c = = λ' γ ( c v) Δt = 1 with = γ (1 β ) Δt β v c Note that the frequency of the source in its rest-frame is: Therefore: f source f obs = c = λ c cδt f γ ( 1 β ) = source = 1 Δt

9 The optical Doppler effect 1+ β f obs = f source 1 β with β = v c If source is approaching 1 à f obs f source, if source and observer are approaching each other 1 β f obs = f source 1 + β If source is receding 1 à f obs f source, if source and observer are receding from each other Questions?

10 Applications in astronomy How far are galaxies away? Is the universe inflating or collapsing? Are there planets outside our solar system? (aka. exoplanets ) Is there dark energy & dark matter?

11 How many stars in this picture? This spiral galaxy probably contains about 200 billion stars (= 200,000,000,000 stars!) There are galaxies that contain ~1000 times more stars than this spiral galaxy! There are several hundred galaxies in this picture! So there are at least 100 trillion stars in this picture!! ~100,000,000,000,000 stars! (The visible universe contains approximately stars: 1,000,000,000,000,000,000,000,000)

12 How far away are these galaxies? How fast are they moving? Where are they going? What holds them together?

13 Edwin Hubble: Use the Doppler effect! If a star is receding, it s light should look reddish (longer wavelength) If a star is approaching, it s light should look bluish (shorter wavelength) v v But how can we measure this? Quantum mechanics comes to the rescue!!

14 Spectra of elements are discrete Hydrogen has very well known lines. Seen through diffraction grating we can distinguish individual, narrow lines. (More about this in the second half of this course!) Now we just have to measure the position of these lines. Each of these hydrogen atoms is like having an accurate clock on a distant star!

15 Spectra of elements are discrete Atoms on earth Atoms on star Measure by how much the lines have shifted due to Doppler effect.

16 Velocity of stars in a galaxy L L Let s investigate the internal working of a galaxy R R longer wavelength lower frequency Which side is moving towards us? a) Left b) Right

17 How fast are the stars at the edges? Δλ = 0.5 nm f λ source = nm obs f γ ( 1 β ) = source fsource 1 β for v << c

18 How fast are the stars at the edges? Δλ = 0.5 nm λ source = nm f obs f source 1 β f = Now we need to convert this to an expression in λ! Which of the following is the correct relation? c λ λ λ obs λsource 1 β A) B) obs c λ source ( 1 β ) C) D) λ λ obs obs E) None of the above λ λ source source c ( 1 β ) ( 1 β )

19 How fast are the stars at the edges? Δλ = 0.5 nm λ obs λ source ( 1 β ) v for v << c λ source = nm Δλ λ source c with Numeric value: λ Δλ source Δλ λ β source λ source v c 0.5nm v c = 226km / 664.5nm Dark matter required for this much centripetal acceleration!!" = &$ # $ # s

20 Hubble and the big bang Edwin Hubble, PNAS March 15, 1929 vol. 15 no

21 The big bang: Far-away objects are receding faster: All space is expanding! Red-shift Far-away stars are receding faster! à All of space is expanding! Distance from earth Next step: Is there dark energy, i.e. is the expansion accelerating?

22 Exoplanets (extra-solar planets) Are there any solar systems (i.e. stars with planets around them) besides our own? Of course there are! (It would be a bit foolish not to assume this, considering that there are ~10 24 suns out there!) But how to prove it?! We typically cannot see the planet directly. (Imagine you try to see something really tiny and dark next to a huge and bright object.) à Most methods focus on the star, instead of the planet.

23 Shadowing due to the planet Brightness Time Brightness The planet shadows off part of the star light as it passes in front of it. Time

24 Doppler shift due to re-coil Doppler shift 1 2 Time [planet years]

25 That s the end of our little excursion to the vast universe. (And all I have on relativistic spacetime) Questions?

26 Relativistic mechanics Previously: Ideas of space and time Simple algebra Next: Ideas of momentum and energy Slightly more involved algebra

27 Momentum The classical definition of the momentum p of a particle with mass m is: p=mu. In absence of external forces the total momentum is conserved (Law of conservation of momentum): n i= 1 p i = const. Due to the velocity addition formula, the definition p=mu is not suitable to obtain conservation of momentum in special relativity!! à Need new definition for relativistic momentum!

28 y Conservation of Momentum y' Frame S moves along x with v = u 1x u 2 m m u' 2 m u 1 u 1x u 1y u' 1 m S If u 1 = -u 2 we find: p tot,before = 0 p tot,after = 0 x S' System S' is moving to the right with the velocity v = u 1x. We will use relativistic velocity transformations here. x'

29 Classical momentum y m u 2 = (-u x,-u y ) p 1, before = m(u x, u y ) p 2, before = m(-u x, -u y ) p tot, before = m(0, 0) m u 1 =(u x,u y ) p 1, after = m(u x, -u y ) p 2, after = m(-u x, u y ) p tot, after = m(0, 0) S à p tot, before = p tot, after

30 Galileo (classical): y' p 1, before = m ( 0, u y ) p 2, before = m (-2u x, -u y ) u' 1 u' 2 m p tot, before = m (-2u x, 0) p 1, after = m ( 0, -u y ) p 2, after = m (-2u x, u y ) S' m p tot, after = m (-2u x, 0) x' à p tot, before = p tot, after

31 Velocity transformation (3D) 2 / 1 ' c v u v u u x x x = ( ) 2 / 1 ' c v u u u x y y = γ ( ) 2 / 1 ' c v u u u x z z = γ Relativistic: Classical: u' x = u x v u' y = u y u' z = u z

32 Lorentz transformation y' u' 2 m Use: u' u' x y = = γ ux 1 u v v / c u x y 2 ( 2 1 u v / c ) x S' u' 1 m Algebra Part of HW#4 x' à p tot, before p tot, after