understand that the lines in the emission and absorption spectra from distant objects show an increase in wavelength from their known values

Understanding Red‑Shift in Stellar Spectra and Estimating Stellar Radii

1. Syllabus Alignment (Cambridge AS & A‑Level Physics 9702)

2. Learning Objectives (What You Should Be Able to Do)

• State why emission/absorption lines from distant objects appear at longer wavelengths than in the laboratory.
• Derive the classical Doppler shift for light from basic kinematics.
• Apply both the non‑relativistic and relativistic Doppler formulas correctly.
• Measure a spectral line, propagate uncertainties, and calculate the radial velocity of a star or galaxy.
• Use Hubble’s law to estimate distances for extragalactic objects.
• Convert apparent magnitude and distance into luminosity (linking work‑energy concepts).
• Determine a star’s radius via the Stefan‑Boltzmann law, including an uncertainty estimate.
• Design a simple spectroscopic experiment, identify systematic and random errors, and suggest realistic improvements.

3. Fundamental Concepts

3.1 Physical Quantities, Units & Uncertainty (AO1 + AO2)

• Wavelength λ: nanometres (nm) or metres (m). 1 nm = 10⁻⁹ m.
• Speed of light c = 3.00 × 10⁸ m s⁻¹.
• Radial velocity v_r: kilometres per second (km s⁻¹) → convert to m s⁻¹ when using SI.
• Hubble constant H₀ ≈ 70 km s⁻¹ Mpc⁻¹ (1 Mpc ≈ 3.09 × 10²² m).
• Uncertainty propagation (first‑order):

  \[

  \sigmaf=\sqrt{\left(\frac{\partial f}{\partial x}\sigmax\right)^2+\left(\frac{\partial f}{\partial y}\sigma_y\right)^2+\dots}

  \]

3.2 Kinematics Refresher (AO1)

For a source moving directly away from the observer, the distance between successive wave‑crests changes at a rate equal to the relative speed v_r. If the source emits at frequency f₀ (period T₀ = 1/f₀), the observed period is

\[

T{\text{obs}} = T0\left(1+\frac{v_r}{c}\right)

\]

Since λ = c T, the observed wavelength becomes

\[

\lambda{\text{obs}} = \lambda0\left(1+\frac{v_r}{c}\right)

\]

This is the classical (non‑relativistic) Doppler shift for light.

3.3 Dynamics – Photon Momentum (AO2)

Photon energy and momentum are related by

\[

E = h f = \frac{h c}{\lambda}, \qquad p = \frac{E}{c} = \frac{h}{\lambda}

\]

A red‑shift (increase in λ) therefore reduces photon momentum, consistent with the source losing kinetic energy in the observer’s frame.

3.4 Superposition & Diffraction Gratings (AO1)

A diffraction grating consists of many equally spaced slits. Constructive interference occurs when the path‑difference equals an integer multiple of the wavelength:

\[

d\sin\theta = m\lambda \quad (m = 0, \pm1, \pm2,\dots)

\]

Measuring the angle θ for a known order m gives λ, which is the basis of spectrographs used in the laboratory and in astronomy.

4. Doppler Shift for Light

4.1 Non‑relativistic Formula (v ≪ c)

\[

\frac{\Delta\lambda}{\lambda0} = \frac{\lambda{\text{obs}}-\lambda0}{\lambda0}= \frac{v_r}{c}

\]

Useful for stars in our Galaxy (v ≲ 300 km s⁻¹ ≈ 10⁻³c).

4.2 Relativistic Formula (v comparable to c)

\[

\frac{\lambda{\text{obs}}}{\lambda0}= \sqrt{\frac{1+vr/c}{1-vr/c}}

\]

Reduces to the non‑relativistic expression when v ≪ c. Required for distant galaxies where v can be a few % of c.

4.3 Comparison with Sound Waves

4.4 Worked Low‑Velocity Example (AO2)

Suppose a nearby star shows Hα (rest λ₀ = 656.28 nm) at 656.45 nm.

\[

\Delta\lambda = 0.17\ \text{nm},\qquad

vr = c\frac{\Delta\lambda}{\lambda0}=3.00\times10^5\frac{0.17}{656.28}=78\ \text{km s}^{-1}

\]

Uncertainty (σ_Δλ = 0.02 nm):

\[

\sigmav = c\frac{\sigma{\Delta\lambda}}{\lambda_0}=3.00\times10^5\frac{0.02}{656.28}\approx 9\ \text{km s}^{-1}

\]

5. Photon Energy and the Red‑Shift

Photon energy:

\[

E = \frac{h c}{\lambda}

\]

Because λ increases, E decreases. For the Hα example above:

\[

E_{\text{rest}} = \frac{6.626\times10^{-34}\times3.00\times10^8}{656.28\times10^{-9}}=3.03\times10^{-19}\ \text{J}

\]

\[

E_{\text{obs}} = \frac{6.626\times10^{-34}\times3.00\times10^8}{656.45\times10^{-9}}=3.02\times10^{-19}\ \text{J}

\]

The tiny energy loss corresponds to the kinetic energy transferred to the expanding Universe (cosmological red‑shift) or to the relative motion (Doppler red‑shift).

6. Practical Spectroscopy – Measuring λ

6.1 Equipment (AO3)

• Medium‑resolution spectrograph (grating or prism) mounted on a stable telescope.
• CCD detector (pixel scale known, e.g. 0.02 nm pixel⁻¹ after calibration).
• Calibration lamp (Hg‑Ar, Ne) with accurately tabulated laboratory wavelengths.
• Computer with spectrum‑analysis software (Gaussian fitting, cross‑correlation).

6.2 Calibration Procedure

1. Take a lamp exposure with the same spectrograph settings as the stellar exposure.
2. Identify at least three well‑separated calibration lines (see Table 6.2).
3. Fit a linear (or low‑order polynomial) relation between pixel number p and wavelength:

   \(\lambda = a p + b\).

4. Record the standard error of the fit (σ_sys) – this is the systematic wavelength uncertainty.

6.3 Measuring a Stellar Line

1. Locate the line of interest (e.g. Hα) in the calibrated spectrum.
2. Fit a Gaussian profile to obtain the line centre \(p{\text{star}}\) and its statistical error \(\sigma{p}\).
3. Convert to wavelength: \(\lambda{\text{obs}} = a p{\text{star}} + b\).
4. Combine uncertainties:

   \[

   \sigma{\lambda}= \sqrt{(a\,\sigma{p})^{2}+ \sigma_{\text{sys}}^{2}}

   \]

6.4 Example Measurement Table

7. From Wavelength Shift to Radial Velocity

Using the non‑relativistic formula (valid for \(v \lesssim 0.01c\)):

\[

v{r}=c\,\frac{\Delta\lambda}{\lambda{0}}

\]

Uncertainty propagation:

\[

\sigma{v}=c\sqrt{\left(\frac{\sigma{\Delta\lambda}}{\lambda_{0}}\right)^{2}

+\left(\frac{\Delta\lambda\,\sigma{\lambda{0}}}{\lambda_{0}^{2}}\right)^{2}}

\]

Because laboratory wavelengths are known to < 0.001 nm, the dominant term is \(\sigma_{\Delta\lambda}\).

8. Distance from Radial Velocity (Extragalactic Objects)

If the object belongs to a galaxy whose recession follows Hubble’s law:

\[

d=\frac{v{r}}{H{0}},\qquad H_{0}=70\ \text{km s}^{-1}\,\text{Mpc}^{-1}

\]

Uncertainty:

\[

\sigma{d}=d\sqrt{\left(\frac{\sigma{v}}{v_{r}}\right)^{2}

+\left(\frac{\sigma{H}}{H{0}}\right)^{2}},\qquad\sigma_{H}\approx5\ \text{km s}^{-1}\,\text{Mpc}^{-1}

\]

9. From Photometry to Luminosity (Link to Work‑Energy)

1. Measure the apparent magnitude \(m\) of the object.
2. Apply the distance modulus:

   \[

   M = m - 5\log_{10}\!\left(\frac{d}{10\ \text{pc}}\right)

   \]

   where \(M\) is the absolute magnitude.

3. Convert \(M\) to luminosity:

   \[

   \frac{L}{L{\odot}} = 10^{(M{\odot}-M)/2.5},\qquad M_{\odot}=4.74

   \]

   and \(L_{\odot}=3.828\times10^{26}\ \text{W}\).

4. Propagate uncertainties through the logarithms using

   \(\sigma{\log x}= \sigmax/(x\ln10)\).

5. Interpretation: \(L = \frac{dE}{dt}\) – the power radiated by the star.

10. Stellar Radius from the Stefan‑Boltzmann Law

Stefan‑Boltzmann law:

\[

L = 4\pi R^{2}\sigma T_{\text{eff}}^{4}

\]

Re‑arranged for radius:

\[

R = \sqrt{\frac{L}{4\pi\sigma T_{\text{eff}}^{4}}}

\]

Uncertainty (independent errors):

\[

\frac{\sigma{R}}{R}= \frac12\sqrt{\left(\frac{\sigma{L}}{L}\right)^{2}

+ \left(4\frac{\sigma{T}}{T{\text{eff}}}\right)^{2}}

\]

Worked Example (including uncertainties)

1. Measured line: Hα λ_obs = 658.00 ± 0.03 nm.
2. Shift: Δλ = 1.72 ± 0.03 nm.
3. Radial velocity:

   \[

   v_{r}=3.00\times10^{5}\frac{1.72}{656.28}=785\ \text{km s}^{-1}

   \]

   \[

   \sigma_{v}=3.00\times10^{5}\frac{0.03}{656.28}\approx14\ \text{km s}^{-1}

   \]

4. Distance (Hubble):

   \[

   d=\frac{785}{70}=11.2\ \text{Mpc}

   \]

   \[

   \sigma_{d}=11.2\sqrt{\left(\frac{14}{785}\right)^{2}+\left(\frac{5}{70}\right)^{2}}\approx1.0\ \text{Mpc}

   \]

5. Apparent magnitude: \(m=12.3\pm0.1\).

   Distance modulus → \(M=-20.3\pm0.2\).

6. Luminosity:

   \[

   L = 10^{(4.74-(-20.3))/2.5}L_{\odot}=2.5\times10^{28}\ \text{W}

   \]

   \(\sigma_{L}\approx0.2L\) (from σ_M).

7. Effective temperature: \(T_{\text{eff}}=8000\pm200\ \text{K}\) (spectral type estimate).
8. Radius:

   \[

   R=\sqrt{\frac{2.5\times10^{28}}{4\pi(5.67\times10^{-8})(8000)^{4}}}=1.2\times10^{9}\ \text{m}=1.7\,R_{\odot}

   \]

   \[

   \frac{\sigma_{R}}{R}= \frac12\sqrt{(0.2)^{2}+(4\times0.025)^{2}}\approx0.12

   \]

   \(\sigma{R}\approx0.2\,R{\odot}\).

11. Experimental Design & Evaluation (AO3)

• Goal: Determine the radial velocity of a distant star (or galaxy) from its Hα absorption line.
• Method outline:

  1. Mount the spectrograph on a stable, thermally equilibrated telescope.
  2. Record a calibration spectrum (Hg‑Ar lamp) before and after the stellar exposure.
  3. Take several 300 s exposures of the target to improve signal‑to‑noise.
  4. Reduce the data: bias subtraction, dark correction, flat‑fielding, and combine the exposures.
  5. Calibrate wavelength using the procedure in §6.2, then fit Gaussian profiles to the Hα line.
  6. Calculate \(v_{r}\) with propagated uncertainties.

• Sources of error

  • Systematic: grating mis‑alignment, temperature‑induced drift of the detector, inaccurate laboratory wavelengths of the calibration lamp.
  • Random: photon (shot) noise, read‑out noise, variable atmospheric seeing.

• Improvements (AO3)

  • Use an iodine absorption cell to provide a simultaneous reference spectrum.
  • Upgrade to an echelle spectrograph (R ≈ 50 000) to resolve blended lines.
  • Apply cross‑correlation with high‑resolution template spectra for sub‑km s⁻¹ precision.
  • Stabilise the spectrograph in a temperature‑controlled enclosure.

12. Key Points to Remember (AO1 + AO2)

• Red‑shift means \(\lambda{\text{obs}} > \lambda{0}\); the shift directly yields the radial component of velocity.
• Use the non‑relativistic Doppler formula for \(v \lesssim 0.01c\); otherwise apply the relativistic expression.
• Accurate wavelength measurement relies on careful calibration and a full uncertainty analysis (systematic + random).
• For extragalactic objects, radial velocity → distance via Hubble’s law; for Galactic stars, the shift gives only the line‑of‑sight speed.
• Luminosity follows from the distance modulus (work‑energy link) and is required for the Stefan‑Boltzmann radius calculation.
• Propagation of uncertainties at each stage is essential for a credible final radius and for meeting AO2/AO3 assessment objectives.