line spectra / spectroscopy - today at...

PHGN324: line spectra / spectroscopyFred Sarazin ([email protected])Physics Department, Colorado School of Mines

Line Spectra / SpectroscopyApplications to astronomy / astrophysics


Line Spectra

• 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.

With d: distance between slits.


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.


• The line spectrum for Hydrogen can completely be explained by solving the Schrödinger equation for the Hydrogen atom.

Quantum mechanics

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

Light source(s)

• Two ways to determine the composition of gas clouds and nebulas

• 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.


Diffraction grating: about 1000 slits per mm

Grating spectrograph


Expectation

Measurement

But wait… there’s more!

• 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 / fs with fs, frequency at the source S

Listener: Relative velocity of the wave front = vL + vSame wavelength, but now different frequency: fL = (vL+v)/ll = v / fs

fL =vL + vv

fS


(Classical) Doppler Effect: Moving Source, Moving Listener

Source: moving at velocity vs. During one cycle: Ts = 1/fs, the wave travels a distance of: vTs = v/fs, while the source travels vsTs=vs/fs.Wavelength (distance between two crests):

In front of the source: l = (v / fs – vs / fs)Behind the source: l = (v / fs + vs / fs)

Listener: Relative velocity of the wave front = vL+vfL = (vL+v)/l with l = (v + vs ) / fs

fL =vL + vvS + v

fS


Application to astronomy (with v0<<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=vr-vs. The measurement is done using the frequency / wavelength shift of e.m. radiation travelling at v=c.

• Other useful formula (in terms of wavelength) using l = c/f

f = (1+ Δvc) f0

With f0 (l0): source frequency (wavelength)f (l): observed frequency (wavelength)Dv: relative radial velocityf = c / l

Note: relative velocity has a sign (approching or moving away)

Δf = f − f0 =Δvcf0 Δλ = λ −λ0 = −

Δvcλ0and

Approching: Df >0 Dl<0 Dv>0 Blue shift

Moving away: Df<0 Dl>0 Dv<0 Red shift

λ = (1− Δvc)λ0


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)

• Source moving away from the receiver: lK = T0(c+Dv)

• Frequency of light: f = c/l (in all reference frames) and f0=1/T0à fK = f0 ( c / (c+Dv) )

• Time dilation: T’K� = gTK à f’K� = fK / gà f’K� = f0 . ( c / g(c+Dv) )

• Which simplifies into: Source and Observer moving away from each other

Dv lK

LightK K�

f = 1−β1+β

f0

β =Δvc

γ =11−β 2

With and


• Source approaching the receiver: lK = T0(c-Dv)

• Frequency of light: f = c/l (in all reference frames) and f0=1/T0à fK = f0 ( c / (c-Dv) )

• Time dilation: T’K� = gTK à f’K� = fK / gà f’K� = f0 . ( c / g(c-Dv) )

• Which simplifies into:

Relativistic Doppler Shift (II)

Source and Observer approaching each other

Dv lK

LightKK�

f = 1+β1−β

f0

β =Δvc

γ =11−β 2

With and


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+β1−β

f0

*Note: the formula reduces to the classical one if Dv<<c


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?

Exercise – relativistic Doppler shift


Redshift z

• Redshift z definition (assuming motion exclusively in radial direction and flat space – Minkowski space)

• 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

1+ z = 1+β1−β

See derivations on the board


Redshift z (useful formula)

z = λ −λ0λ0

1+ z = λλ0

1+ z = f0f

z = f0 − ff

Based on wavelength Based on frequency

Calculation of redshift z

See derivations on the board


Exercise – Quasar redshift

Quasar redshiftThe 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?

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