Line Spectra / Spectroscopy Applications to astronomy / astrophysics

Line Spectra / Spectroscopy Applications to astronomy / astrophysics

Line Spectra With d: distance between slits. It is observed that chemical elements produce unique colors when burned (with a flame) or excited (with an electrical discharge) Diffraction creates a line spectrum pattern of light bands and dark areas on the screen. The line spectrum serves as a fingerprint of the gas that allows for unique identification of chemical elements and material composition.

Balmer Series for Hydrogen In 1885, Johann Balmer (a swiss schoolteacher) finds an empirical formula for wavelength of the visible hydrogen line spectra in nm: nm (where k = 3,4,5 ) à Underlying order/quantification not understood at the time

Rydberg equation As more spectral lines are discovered, a more general empirical equation appears: the Rydberg equation Rydberg constant (for Hydrogen)

Exercise Line spectra Find the Balmer formula from the Rydberg equation. Determine a formula for the Lyman series and the Pashen series.

Quantum mechanics The line spectrum for Hydrogen can completely be explained by solving the Schrödinger equation for the Hydrogen atom. With V(r), the electrostatic potential

Other elements

Absorption vs Emission spectrum

The line spectra of stars (I) Absorption spectrum of stars: Inner, dense layers of the star produce a continuous (blackbody) spectrum Cooler surface layers absorb light at specific wavelengths / frequencies

The line spectra of stars (II): assessing how old is a star Metal-poor star (very old star) Metal-rich star (relatively young star) The Sun is a metal-rich star

The line spectra of stars (III): typical characteristics Emission and absorption spectra of stars as a function of temperature Remember that emission spectrum shape depends on temperature (blackbody radiation), hence emission spectrum peaks at different wavelengths / frequencies. blue star red star

Composition of gas clouds and nebula Two ways to determine the composition of gas clouds and nebulas Light source(s) Emission spectrum from a planetary nebula (H, He, O, Ne) Planetary nebula: expanding shell of ionized gas ejected from old red giant stars late in their lives.

Grating spectrograph Diffraction grating: about 1000 slits per mm

But wait there s more! Measurement Expectation Doppler shift The displacement of the spectral lines informs us on the relative motion of the astrophysical object with respect to us! Red-shift: the object is moving away from us Blue-shift: the object is moving towards us

Doppler shift of spectra lines allows for precise measurements (I)

Doppler shift of spectra lines allows for precise measurements (II)

(Classical) Doppler Effect: Fixed Source, Moving Listener Source: Wavelength: l = v / f s with f s, frequency at the source S Listener: Relative velocity of the wave front = v L + v Same wavelength, but now different frequency: f L = (v L +v)/l l = v / f s f L = v L + v v f S

(Classical) Doppler Effect: Moving Source, Moving Listener Source: moving at velocity v s. During one cycle: T s = 1/f s, the wave travels a distance of: vt s = v/f s, while the source travels v s T s =v s /f s. Wavelength (distance between two crests): In front of the source: l = (v / f s v s / f s ) Behind the source: l = (v / f s + v s / f s ) Listener: Relative velocity of the wave front = v L +v f L = (v L +v)/l with l = (v + v s ) / f s f L = v L + v v S + v f S

Application to astronomy (with v 0 <<c) (a) Police radar can measure only the radial part of your velocity (Vr) as you drive down the highway, not your true velocity along the pavement (V). That is why police using radar should never park far from the highway. This police car is poorly placed to make a good measurement. (b) From Earth, astronomers can use the Doppler effect to measure the radial velocity (Vr) of a star, but they cannot measure its true velocity, V, through space.

Application to astronomy Dv<<c In astronomy, the Doppler shift allows for the measurement of the relative radial velocity Dv=v r -v s. The measurement is done using the frequency / wavelength shift of e.m. radiation travelling at v=c. f = (1+ Δv c ) f 0 λ = (1 Δv c )λ 0 With Other useful formula (in terms of wavelength) using l = c/f Δf = f f 0 = Δv c f 0 Note: and f 0 (l 0 ): source frequency (wavelength) f (l): observed frequency (wavelength) Dv: relative radial velocity f = c / l relative velocity has a sign (approching or moving away) Δλ = λ λ 0 = Δv c λ 0 Approching: Df >0 Dl<0 Dv>0 Blue shift Moving away: Df<0 Dl>0 Dv<0 Red shift

Classical Doppler shift exercise Suppose the laboratory wavelength of a certain spectral line is 600.00nm and the line is observed in a star spectrum at a wavelength of 600.10nm*. 1. Is the star moving towards us or away from us? 1. What is the radial velocity of the star with respect to Earth? * Notice that you need a rather precise instrument to measure this wavelength shift.

Relativistic Doppler Shift (I) Dv K l K Light K Source moving away from the receiver: l K = T 0 (c+dv) Frequency of light: f = c/l (in all reference frames) and f 0 =1/T 0 à f K = f 0 ( c / (c+dv) ) Time dilation: T K = gt K à f K = f K / g à f K = f 0. ( c / g(c+dv) ) With β = Δv c and γ = 1 1 β 2 Which simplifies into: Source and Observer f = 1 β 1+ β f 0 moving away from each other

K Light Dv lk Relativistic Doppler Shift (II) K Source approaching the receiver: l K = T 0 (c-dv) Frequency of light: f = c/l (in all reference frames) and f 0 =1/T 0 à f K = f 0 ( c / (c-dv) ) Time dilation: T K = gt K à f K = f K / g à f K = f 0. ( c / g(c-dv) ) Which simplifies into: f = 1+ β 1 β f 0 With β = Δv c and γ = Source and Observer approaching each other 1 1 β 2

Relativistic Doppler Shift (III) Assuming b = Dv/c including sign*: b = Dv/c (source and observer approaching) b = -Dv/c (source and observer moving away) f = 1+ β 1 β f 0 *Note: the formula reduces to the classical one if Dv<<c

Exercise relativistic Doppler shift A long time ago, in a galaxy far away Death Star destroying the Alderaan planet: A fleet from the planet Alderaan is flying towards the death star at 0.3c (with respect to the death star), when the death ray beam destroys Alderaan. The death ray (a powerful laser beam) appears green (l=550nm) to the stormtroopers on board the death star. What is the color of the death ray beam seen by the pilots of the intercepting fleet?

Redshift z Redshift z definition (assuming motion exclusively in radial direction and flat space Minkowski space) 1+ z = 1+ β 1 β Distant objects such as quasars have strongly red-shifted line spectra. This is due to the fast expansion of the Universe and is relevant to Cosmology. In fact, all the far-away astrophysical objects display red-shifted lines, hence it is useful to define z as a positive quantity when objects are moving away from us. There is a clear connection between the relativistic Doppler effect formula and the redshift one. HOWEVER, the sign of b is the opposite. Relativistic Doppler effect: b>0 source and observer approaching Redshift: b>0 source and observer moving apart See derivations on the board

Redshift z (useful formula) Calculation of redshift z Based on wavelength z = λ λ 0 λ 0 Based on frequency z = f 0 f f 1+ z = λ λ 0 1+ z = f 0 f See derivations on the board

Exercise Quasar redshift Quasar redshift The most distant objects known in the Universe are objects called quasars. Their light is highly red-shifted, a fact that is usually taken to mean that quasars are receding from us rapidly. Quasars have been observed with a redshift of z=4.9. 1. What is their relative velocity b=dv/c with respect to us? 2. Can you guess that the redshift we are talking about here is different from the redshift measured at the local galactic scale?