The VLT (Chile) credit: Waldmann L9: The Doppler velocity UCL Certificate of astronomy Dr. Ingo Waldmann
This week in the news
Detecting exoplanets through their motion We have seen that the presence of a planet moves the centre of mass away from the star. The planet orbits this centre of mass (CoM). The star also orbits this CoM although it will be located very close to it: wobble instead of orbital motion CoM Exoplanets can therefore be detected by the small perturbation of the star. The more massive/closer the planet is, the stronger this perturbation (and the easier the detection)
How can we detect this motion? The planet is often too faint and hidden behind the glare of the parent star Position Observe the star - for years - and measure the relative position with respect to distant stars along the line of sight. The amplitude and period can be used to determine the mass and distance of planet(s). Velocity Somehow measure the velocity of the star and look for periodic motion. The period and velocity range of this motion can be used to determine the mass and distance of perturbing planet(s). Velocity Time
How to measure velocities The only information we can get from stars comes from their light. Light has a built-in clock given by the wavelength/frequency If stellar light is passed through a prism, we decompose this light into its colours: the energy flux is a function of wavelength F(λ) Blackbody + Absorption Cooler Envelope Hot core
An example: The Sun s Fraunhofer lines 13,000 Å Solar Spectrum wavelength 2,960 Å N.A.Sharp, NOAO/NSO/Kitt Peak FTS/AURA/NSF
An example: The sun
The discrete nature of stellar lines The absorption lines seen in stellar spectra correspond to very well defined transitions in atoms/molecules. The wavelength/frequency of these transitions can be determined with high accuracy using quantum mechanics (QM) An example: The Hydrogen atom: Solving the problem of a proton and an electron as a bound system using QM (Schrödinger s equation) gives a set of discrete energy levels determined by an integer number n: E(n) = -13.6eV/n 2, where n=1,2,3, The transition between any of these two levels implies energy will be emitted/absorbed (photons) 13.6eV photon
The discrete nature of stellar lines Hα photon ΔE=E 3 -E 2 =1.89eV ΔE=hν=hc/λ λ= 656nm
Spectrum of Vega (α Lyr) K. Andrew (EIU)
The Doppler Effect The extra piece of information we need in order to use these absorption lines as a speedometer is the Doppler Effect: The measured wavelength/frequency of light depend on the relative motion between source (star) and detector (us). This behaviour happens with all sorts of waves, including sound waves. It makes the sound of a siren higher in frequency as the ambulance gets closer to us, and lower in frequency as it moves away from us. Simple kinematic argument
The Doppler Effect Consider first the motion of a wave between a source and a detector which stay at rest. Source (rest-frame) Period: τ 0 = t 2 -t 1 Detector (observer-frame) Period: τ=t 4 -t 3 =τ 0 τ = τ 0 λ(=cτ) = λ 0
The Doppler Effect Now the detector moves away from the source at velocity v: Source (rest-frame) Period: τ 0 = t 2 -t 1 Detector (observer-frame) Period: τ=t 5 -t 3 = τ 0 + τ 0 v/c τ = τ 0 (1+v/c) λ(=cτ) = λ 0 (1+v/c)
The Doppler Effect If the detector moves away from the source, we have to wait for a bit longer until we observe the second peak of the wave: longer period, longer wavelength (redshift) If the detector moves towards the source (negative value of v), the observed period (and wavelength) are shorter (blueshift) This velocity is actually the velocity towards/away from source: line-ofsight velocity. The transverse velocity does not change wavelengths/frequencies Zero Doppler Effect Maximum Doppler Effect
The Doppler Effect The expression for the Doppler effect: = 0 1+ v c λ = λ 0 (1+v /c) is actually NOT a shift (translation) towards red/blue but a stretch/compression of the spectrum The effect of this change is very small: Δλ/λ 0 =(λ-λ 0 )/λ 0 = v /c = ( 0) 0 = v c i.e. the fractional change of wavelength depends on the ratio between the projected velocity along the line of sight and the speed of light (300,000 km/s)
The Doppler Effect Velocities of stars in our Galaxy amount up to a few times 100km/s, which results in optical wavelength changes ~4-5 Å 1Å=10-10 m, visible light has wavelength ~4000Å (blue) - 7000Å(red) Extra-solar planets may exert a wobbly motion of order a few m/s, which implies fractional wavelength changes in the optical ~0.1 Å
Galaxies and the Doppler Effect IC 3509 Δλ/λ 0 =0.73 D PM = 8.5 10 9 lyr J033232.73-275102.5 Δλ/λ 0 =0.007 D PM = 104 10 6 lyr SDSS 13.4Gyr Age of Universe 7 Gyr GOODS/CDFS ESO-VLT/FORS2
Cosmological red-shift In cosmology the standard distance metric is the cosmological red-shift z z = observed emitted 1 v c red-shift of Andromeda galaxy is z = - 0.001 (it is actually coming towards us)! red-shift of the Cosmic Microwave Background (CMB) is z ~ 1100 Padmanabhan (2007)
Aims and Objectives A first introduction to the main methods to discover extra-solar planets Mechanics and gravitation: orbits of two-body systems Measuring motion with light Doppler effect and spectroscopy: The relative motion between source and observer changes the period (τ), wavelength (λ=cτ) or frequency (ν=c/λ=1/τ) of light The relative change in wavelength is Δλ/λ 0 / = v 0 = /c v/c Velocities ~few 100 km/s can be measured with 5 Å resolution at optical wavelengths Velocity changes imparted by exo-planets on their host stars are much lower ~ few m/s and require much higher spectral resolution (0.1 Å)