Doppler shift, redshift and the universe flashcards

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Doppler effect
The change in observed frequency and wavelength of a wave when the source and observer move relative to each other.
Redshift
An increase in the observed wavelength (decrease in frequency) of radiation from a source moving away from the observer.
Blueshift
A decrease in the observed wavelength (increase in frequency) of radiation from a source moving towards the observer.
Hubble's law
The recession speed of a distant galaxy is proportional to its distance: v = H₀d.
Critical density
The mean density of the universe at which gravity would just stop the expansion after an infinite time.
Dark matter
Matter that emits and absorbs no detectable electromagnetic radiation; it is detected only by its gravitational effects.
Redshift equation
z = Δλ/λ ≈ Δf/f ≈ v/c (valid for v ≪ c). (Yes)
Hubble's law as an equation
v = H₀d. Gradient of a v–d graph = H₀. (Yes)
Age of the universe estimate
t ≈ 1/H₀ (H₀ in s⁻¹), assuming a constant expansion rate. (No — derive it)
Orbital speed of a star at radius r around mass M
v = √(GM/r), from GMm/r² = mv²/r. So M = v²r/G. (No — derive it)
Wavefronts from a moving source
Circles centred on where the source was at each emission: bunched ahead (shorter λ), spread out behind (longer λ).
Shape of a Hubble graph (v against d)
A straight line through the origin; gradient = H₀. Points scatter because distances are uncertain and galaxies have their own motions.
Galaxy rotation curve: predicted vs observed
Predicted from visible mass: speed falls with distance beyond the bright centre. Observed: speed stays roughly constant (flat).
Size of the universe against time: closed, flat, open
Closed: expands, stops, collapses. Flat: expands for ever, rate → 0. Open: expands for ever. Denser models are younger.
Redshifted absorption spectrum
The same pattern of lines, all moved to longer wavelengths; each line shifts by Δλ = zλ, so red lines move further than blue ones.
Why does a moving source change the wavelength received but not the wave speed?
Wave speed is set by the medium (or is c). The source moves between emissions, so only the spacing of the wavefronts changes.
Observer moving towards a stationary source
Wavelength in the medium is unchanged, but the observer meets wavefronts more often: higher frequency.
Why does Hubble's law suggest the universe is expanding?
All distant galaxies recede, with speed proportional to distance — the pattern every observer sees in a uniformly expanding universe.
Why is 1/H₀ only an estimate of the age?
It assumes constant expansion. Gravity has slowed the expansion, so the true age is less; H₀ itself is also uncertain.
Why is the fate of the universe uncertain?
It depends on the mean density compared with the critical density, and the amount of dark matter (hence the density) is uncertain.
How do rotation curves show dark matter?
v = √(GM/r): a flat curve needs M ∝ r far beyond the visible stars, so there is unseen mass.
Hα (656.3 nm in the lab) is observed at 658.1 nm. Speed and direction?
v = 3.00 × 10⁸ × 1.8 ÷ 656.3 = 8.2 × 10⁵ m s⁻¹, moving away (redshift). (Yes)
A galaxy has z = 0.020. Distance if H₀ = 70 km s⁻¹ Mpc⁻¹?
v = 0.020 × 3.00 × 10⁵ = 6000 km s⁻¹; d = 6000 ÷ 70 = 86 Mpc. (Yes)
Age of the universe from 1/H₀ if H₀ = 65 km s⁻¹ Mpc⁻¹? (1 Mpc = 3.09 × 10²² m)
H₀ = 65 × 10³ ÷ 3.09 × 10²² = 2.10 × 10⁻¹⁸ s⁻¹; 1/H₀ = 4.75 × 10¹⁷ s ≈ 1.5 × 10¹⁰ years.
Gas orbits 10 kpc from a galaxy's centre at 200 km s⁻¹. Mass inside the orbit? (1 kpc = 3.09 × 10¹⁹ m)
M = v²r/G = (2.0 × 10⁵)² × 3.09 × 10²⁰ ÷ 6.67 × 10⁻¹¹ = 1.9 × 10⁴¹ kg.
Opposite edges of a galaxy show a line 0.80 nm apart (λ ≈ 600 nm). Rotation speed?
Half the difference: 0.40 nm. v = 3.00 × 10⁸ × 0.40 ÷ 600 = 2.0 × 10⁵ m s⁻¹.
Order of subtraction for Δλ
Δλ = observed − laboratory. Positive: redshift (away). Negative: blueshift (towards). Divide by the laboratory λ.
Units in v = H₀d
H₀ in km s⁻¹ Mpc⁻¹ needs v in km s⁻¹ and gives d in Mpc. For SI, convert H₀ to s⁻¹ first.
Does a larger H₀ mean an older universe?
No — a larger H₀ is a faster expansion, so 1/H₀ is smaller: a younger universe.
When does z = v/c fail?
When v is not much less than c (z above about 0.1). Say the approximation is no longer valid.

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