Cambridge International AS & A Level Physics (9702) – Stellar Radii from Doppler Red‑shift
1. Syllabus map – where this topic fits
| Syllabus block | Relevant topics (numbers) | How the red‑shift material links |
|---|
| 1–5 : Quantities, Kinematics, Dynamics, Forces, Energy | 1, 2, 3, 4, 5 | - Vector nature of velocity – only the radial component vr enters the Doppler formula.
- Non‑relativistic Doppler shift Δλ/λ = v/c derived from basic kinematics.
- Orbital kinetic energy ½ M v2 links the measured radial speed to the energy of the binary system (work‑energy principle).
|
| 6–8 : Deformation, Waves, Superposition | 6, 7, 8 | - Light is an electromagnetic wave: c = fλ.
- The Doppler effect changes λ and f because successive wave‑fronts are stretched or compressed.
- In a spectroscopic binary the observed line profile is the superposition of two Doppler‑shifted components (one from each star). The resulting line splitting is a direct illustration of wave superposition.
- Instrumental spectral resolution (Δλ/λ ≈ 10⁻⁴) is itself a consequence of the finite width of interference fringes – an application of wave superposition.
|
| 9–11 : Electricity, DC circuits, Particle physics | 9, 10, 11 | - The empirical mass–radius relation for main‑sequence stars is ultimately rooted in stellar‑structure physics, which depends on nuclear processes (hydrogen fusion, binding energy).
- Thus the relation provides a bridge to the particle‑physics extensions (topic 23 – nuclear binding energy).
|
| 12–25 : Gravitation, Thermodynamics, Oscillations, Fields, AC, Quantum, Nuclear, Medical, Astronomy | 13, 17, 22, 24, 25 | - Gravitation (13) – Kepler’s laws give the link between orbital size, period and the total mass of the binary.
- Oscillations (17) – For a circular orbit the radial motion of each star is simple harmonic; the velocity curve is sinusoidal.
- Quantum (22) – Identification of spectral lines uses E = hf; e.g. the Balmer series (H‑α, H‑β …) provides the laboratory wavelength λ₀.
- Medical physics (24) – Doppler ultrasound measures blood‑flow speed with the same Δf/f = −v/c relation; a short example is given in §8.
- Astronomy (25) – Determining stellar radii from binary data is a classic application of hydrostatic equilibrium and energy generation in stars.
|
2. Quick recap boxes
Quantities you need
- Radial velocity, vr – component of the star’s velocity along the line of sight (positive = receding).
- Speed of light, c – 3.00 × 10⁸ m s⁻¹ (constant in vacuum).
- Wavelength, λ and frequency, f – related by c = fλ.
- Period, P – orbital period of the binary (s).
- Semi‑major axis, a – distance from a star to the centre of mass.
- Inclination, i – angle between the orbital plane and the plane of the sky (i = 90° for an edge‑on eclipse).
- Eccentricity, e – shape of the orbit (e = 0 for a circle).
Wave nature of light (topics 6–8)
Light propagates as an electromagnetic wave with speed c, frequency f and wavelength λ. When source and observer move relative to each other the spacing of successive wave‑fronts changes – this is the Doppler effect.
For sound a medium is required; for light there is none, so only the relative motion matters.
3. The Doppler shift – from fundamentals to formulae
3.1 Non‑relativistic derivation (v ≪ c)
Consider a source moving directly away from the observer with speed v. In the source’s rest frame successive crests are emitted every period 1/f₀, i.e. with spacing λ₀ = c/f₀.
During the time Δt = λ₀/c between two crests the source travels an extra distance vΔt. The observed spacing therefore becomes
\[
\lambda{\text{obs}} = \lambda0 + v\Delta t = \lambda_0\left(1+\frac{v}{c}\right).
\]
Hence the fractional wavelength shift is
\[
\boxed{\frac{\Delta\lambda}{\lambda0}= \frac{\lambda{\text{obs}}-\lambda0}{\lambda0}= \frac{v}{c}}.
\]
Because c = fλ, the corresponding fractional frequency shift is
\[
\boxed{\frac{\Delta f}{f_0}= -\,\frac{v}{c}}.
\]
3.2 Relativistic correction (v ≈ 0.05 c or larger)
\[
\boxed{\frac{\lambda{\text{obs}}}{\lambda0}= \sqrt{\frac{1+v/c}{\,1-v/c\,}}}
\qquad\text{or}\qquad
\boxed{\frac{f{\text{obs}}}{f0}= \sqrt{\frac{1-v/c}{\,1+v/c\,}}}.
\]
For A‑level work the non‑relativistic expression is sufficient unless the question explicitly states “high‑speed” or gives v ≥ 0.1 c.
3.3 Sign convention
- v > 0 → source receding → red‑shift (Δλ > 0, Δf < 0).
- v < 0 → source approaching → blue‑shift (Δλ < 0, Δf > 0).
4. Measuring radial velocity from a spectral line
- Identify a laboratory (rest) wavelength λ₀ of a strong, isolated line (e.g. H‑α = 656.28 nm). The value comes from photon‑energy = hf and the Balmer series.
- Measure the observed wavelength λobs from the stellar spectrum (spectrograph or calibrated CCD). The line profile may be the superposition of two shifted components in a binary.
- Apply the non‑relativistic formula:
\[
v = c\,\frac{\lambda{\text{obs}}-\lambda0}{\lambda_0}.
\]
Typical spectrograph resolutions give Δλ/λ ≈ 10⁻⁴, corresponding to a few km s⁻¹ precision.
Example (single star)
Observed H‑α line: λobs = 656.45 nm.
\[
\Delta\lambda = 656.45-656.28 = 0.17\text{ nm}
\]
\[
v = (3.00\times10^{8}\;\text{m s}^{-1})\frac{0.17}{656.28}=7.8\times10^{4}\;\text{m s}^{-1}=78\;\text{km s}^{-1}.
\]
The star is receding (red‑shift).
5. Spectroscopic binaries – linking radial velocity to orbital parameters
5.1 Velocity curve from superposed lines
In a double‑lined binary the observed spectrum is the sum of two Doppler‑shifted line profiles. By fitting two Gaussian (or Voigt) components one obtains the individual radial velocities as a function of time.
The resulting radial‑velocity curve is sinusoidal for a circular orbit (simple harmonic motion):
\[
v_r(t)=K\;\sin\!\bigl(2\pi t/P+\phi\bigr),
\]
where K is the semi‑amplitude.
5.2 From K to orbital size (topic 17 – oscillations)
For a circular orbit the radial motion is simple harmonic, so the amplitude K relates directly to the orbital radius a₁:
\[
\boxed{K = \frac{2\pi a_1\sin i}{P}} \qquad (e=0).
\]
For an eccentric orbit the denominator gains a factor √(1 − e²).
5.3 Kepler’s third law (topic 13)
\[
\boxed{\frac{(a1+a2)^3}{P^2}= \frac{G(M1+M2)}{4\pi^{2}}}.
\]
5.4 Mass function (topic 22 – quantum link via photon energies)
\[
\boxed{f(M)=\frac{(M2\sin i)^3}{(M1+M_2)^2}= \frac{K^{3}P}{2\pi G}\,(1-e^{2})^{3/2}}.
\]
The function is derived solely from the observed Doppler shift (Δλ/λ) and the period, linking spectroscopy (quantum) to gravitation.
5.5 Inclination and eclipses
- For an eclipsing system i ≈ 90° → sin i ≈ 1, simplifying the equations.
- If the system is not eclipsing, sin i must be retained; the mass function then yields a *minimum* mass for the unseen companion.
6. From mass to radius – stellar radius estimate
For main‑sequence stars the empirical mass–radius relation (topic 9–11) is
\[
\boxed{\frac{R}{R{\odot}}\;\approx\;\left(\frac{M}{M{\odot}}\right)^{0.8}}.
\]
This relation is a consequence of stellar‑structure physics: the balance between gravitational pressure (topic 13) and the energy generated by hydrogen fusion (topic 23 – nuclear binding). Thus the radius estimate is firmly linked to the particle‑physics extension of the syllabus.
7. Worked example – full chain from spectrum to radius
- Observed line – H‑α shows two components; the primary’s component yields a semi‑amplitude K = 45 km s⁻¹.
- Orbital period – Light‑curve analysis gives P = 3.2 days = 2.77 × 10⁵ s.
- Inclination – Total eclipses → i = 87° → sin i ≈ 0.998.
- Eccentricity – Velocity curve symmetric → e ≈ 0.
- Primary’s orbital radius (using K‑formula, topic 17):
\[
a_1 = \frac{K\,P}{2\pi\sin i}
=\frac{(45\times10^{3}\,\text{m s}^{-1})(2.77\times10^{5}\,\text{s})}{2\pi(0.998)}
\approx 6.2\times10^{9}\,\text{m}=6.2\times10^{6}\,\text{km}.
\]
- Mass function (topic 22):
\[
f(M)=\frac{K^{3}P}{2\pi G}
=\frac{(45\times10^{3})^{3}(2.77\times10^{5})}{2\pi(6.67\times10^{-11})}
\approx 0.13\,M_{\odot}.
\]
- Assuming the secondary is a G‑type main‑sequence star (≈ 1.0 M⊙), solving the mass‑function gives M₁ ≈ 1.2 M⊙.
- Radius of the primary (mass–radius relation, topic 9–11, linked to nuclear fusion):
\[
R1 = R{\odot}\,(1.2)^{0.8}\approx1.15\,R_{\odot}.
\]
8. Medical‑physics analogue – Doppler ultrasound
In clinical ultrasound the same fractional frequency shift is used:
\[
\frac{\Delta f}{f0}= -\frac{v{\text{blood}}}{c_{\text{sound}}},
\]
where csound ≈ 1540 m s⁻¹ in tissue. By measuring Δf the blood‑flow speed vblood is obtained, illustrating that the physics of red‑shift is universal across electromagnetic and acoustic waves.
9. Summary of key equations
| Quantity | Expression | When to use |
|---|
| Fractional wavelength shift | \(\displaystyle \frac{\Delta\lambda}{\lambda_0}= \frac{v}{c}\) | v ≪ c (most stellar applications) |
| Fractional frequency shift | \(\displaystyle \frac{\Delta f}{f_0}= -\frac{v}{c}\) | Same limit as above |
| Radial velocity from a line | \(\displaystyle v = c\,\frac{\lambda{\text{obs}}-\lambda0}{\lambda_0}\) | Direct measurement of a single‑lined star |
| Semi‑amplitude of velocity curve | \(\displaystyle K = \frac{2\pi a_1\sin i}{P\sqrt{1-e^{2}}}\) | Spectroscopic binary (circular: √ term = 1) |
| Kepler’s third law (binary) | \(\displaystyle \frac{(a1+a2)^3}{P^2}= \frac{G(M1+M2)}{4\pi^{2}}\) | Relates masses and orbital size |
| Mass function | \(\displaystyle f(M)=\frac{K^{3}P}{2\pi G}(1-e^{2})^{3/2}\) | Derives component masses from spectroscopy |
| Mass–radius relation (main‑sequence) | \(\displaystyle \frac{R}{R{\odot}}\approx\left(\frac{M}{M{\odot}}\right)^{0.8}\) | Estimate stellar radius after obtaining M |
| Relativistic Doppler shift | \(\displaystyle \frac{\lambda{\text{obs}}}{\lambda0}= \sqrt{\frac{1+v/c}{1-v/c}}\) | v ≳ 0.05 c or when explicitly required |
10. Practical tips for A‑level examinations
- State clearly whether the observed shift is a red‑shift or a blue‑shift before substituting numbers.
- Check the magnitude of v. If v > 0.03 c, write down the relativistic formula and justify its use.
- Convert all periods to seconds before using Kepler’s law.
- Remember the \(\sin i\) factor – for eclipsing binaries set \(\sin i\approx1\); otherwise keep it.
- When wavelengths are given in Ångström, convert to metres (1 Å = 10⁻¹⁰ m) before using the Doppler equation.
- Show a brief line of reasoning when you invoke the mass–radius relation; the examiner awards marks for the correct conceptual link to stellar structure.
- If a question mentions “spectroscopic binary”, comment on the superposition of two shifted lines – this demonstrates understanding of wave superposition (topic 8).
- For a medical‑physics style question, replace c with the speed of sound in tissue (≈ 1540 m s⁻¹) and use the same Δf/f relation.
11. Checklist – does your answer cover the syllabus?
- Identify the relevant laboratory wavelength (quantum, topic 22).
- Calculate Δλ/λ and obtain v = c Δλ/λ (kinematics, topics 1‑5).
- State the sign convention (red‑shift vs blue‑shift).
- For a binary, extract K, P, e, i from the velocity curve (oscillations, topic 17).
- Apply Kepler’s law and the mass function (gravitation, topic 13).
- Use the mass–radius empirical law and comment on its nuclear‑physics origin (topics 9‑11, 23).
- Optional: discuss superposition of spectral lines and instrumental resolution (waves, topic 8).
- Optional: compare with a Doppler‑ultrasound example (medical physics, topic 24).