Doppler effect (A-level only)

AQA·Physics·3.9.3.1

Doppler shift of electromagnetic radiation

When a source of electromagnetic radiation and an observer move relative to each other along their line of sight, the observer detects a different frequency and wavelength from those emitted. This change is called the Doppler shift.

If they move towards each other, successive waves reach the observer more frequently. The observed frequency is higher and the observed wavelength is shorter. If they move apart, the waves arrive less frequently: the observed frequency is lower and the observed wavelength is longer. These relationships apply to both optical and radio radiation.

Doppler shift at low radial speeds

The radial speed is the component of relative velocity along the line of sight. For the relationships below, define $$v$$ as positive for motion towards the observer and negative for motion away. When the speed is much smaller than the speed of light, $$v \ll c$$, the fractional frequency shift is approximately:

Δff≈vc\frac{\Delta f}{f} \approx \frac{v}{c}

The corresponding fractional wavelength shift, called redshift and denoted by $$z$$, is:

z=Δλλ≈−vcz = \frac{\Delta\lambda}{\lambda} \approx -\frac{v}{c}

Here $$f$$ and $$\lambda$$ are the emitted frequency and wavelength, and $$c$$ is the speed of light. Each change is the observed value minus the emitted value. Thus, approach gives a positive frequency shift but a negative wavelength shift. Recession gives a negative frequency shift and a positive redshift.

Measuring redshift from spectral features

A spectral feature has a known rest wavelength: the wavelength measured when there is no relative radial motion between its source and the observer. Comparing that wavelength with the feature’s observed wavelength gives the redshift:

z=λobserved−λrestλrestz = \frac{\lambda_{\mathrm{observed}}-\lambda_{\mathrm{rest}}}{\lambda_{\mathrm{rest}}}

This comparison can be made for spectral features from galaxies and quasars. A positive $$z$$ means the observed wavelength is longer than the rest wavelength; a negative $$z$$ means it is shorter.

The approximation $$z \approx -v/c$$ can then be used to estimate radial velocity only when the relevant speed is much smaller than $$c$$. It must not be treated as an exact relationship for large redshifts.

Doppler calculations for binary stars

As two stars orbit their common centre of mass, the component of each star’s velocity along the line of sight changes. Each star’s spectral lines therefore shift as it moves towards or away from the observer. These shifts can be used to find the stars’ orbital speeds and radii, and their mass ratio.

From spectral shift to line-of-sight velocity

For speeds much smaller than the speed of light, the fractional shift of a spectral line gives the star’s line-of-sight velocity:

Δλλ0≈uc≈−Δff0\frac{\Delta\lambda}{\lambda_0}\approx\frac{u}{c}\approx-\frac{\Delta f}{f_0}

Here, $$\lambda_0$$ and $$f_0$$ are the wavelength and frequency of the line when the source is at rest relative to the observer. The shifts are $$\Delta\lambda=\lambda-\lambda_0$$ and $$\Delta f=f-f_0$$; $$c$$ is the speed of light.

In this convention, $$u$$ is positive for motion away from the observer. A line shifted to a longer wavelength therefore gives a positive velocity; a line shifted to a shorter wavelength gives a negative velocity. Frequency shifts have the opposite sign.

Orbital speeds and the system’s motion

The measured velocity includes both the motion of the binary system as a whole and the star’s orbital motion. For a circular orbit viewed edge-on, the star’s orbital velocity is sometimes entirely along the line of sight. Its maximum line-of-sight orbital speed is then equal to its orbital speed.

For either star, the midpoint between its greatest and least measured line-of-sight velocities gives the system’s line-of-sight velocity. Half the difference gives the star’s orbital speed:

usystem=umax⁡+umin⁡2,v=umax⁡−umin⁡2u_{\text{system}}=\frac{u_{\max}+u_{\min}}{2},\qquad v=\frac{u_{\max}-u_{\min}}{2}

The two stars move in opposite directions about their common centre of mass. Their orbital contributions to the line-of-sight velocity are therefore opposite, although the motion of the system as a whole can shift both measured velocities in the same direction.

Orbital radii and mass ratio

The stars have the same orbital period $$T$$. In one period, a star travels once around a circle of circumference $$2\pi r$$ at orbital speed $$v$$. Its orbital radius is therefore:

r=vT2πr=\frac{vT}{2\pi}

Apply this relationship separately to each star, using its own orbital speed. The stars’ masses and orbital speeds also satisfy:

m1v1=m2v2,m1m2=v2v1m_1v_1=m_2v_2,\qquad \frac{m_1}{m_2}=\frac{v_2}{v_1}

The more massive star moves more slowly around the centre of mass. These speeds give the mass ratio, not the individual masses by themselves.

Binary-star calculation

Suppose a circular binary is viewed edge-on. Doppler measurements give star 1 line-of-sight velocities ranging from $$-30\ \mathrm{km,s^{-1}}$$ to $$50\ \mathrm{km,s^{-1}}$$. The midpoint is $$10\ \mathrm{km,s^{-1}}$$, so this is the system’s velocity away from the observer. Half the range gives star 1 an orbital speed of $$40\ \mathrm{km,s^{-1}}$$.

Suppose star 2’s measured velocities range from $$-10\ \mathrm{km,s^{-1}}$$ to $$30\ \mathrm{km,s^{-1}}$$. They have the same midpoint, $$10\ \mathrm{km,s^{-1}}$$, and give star 2 an orbital speed of $$20\ \mathrm{km,s^{-1}}$$.

If the common period is $$10$$ days, or $$864{,}000\ \mathrm{s}$$, their orbital radii are:

r1=(40,000)(864,000)2π≈5.5×109 m,r2=(20,000)(864,000)2π≈2.8×109 mr_1=\frac{(40{,}000)(864{,}000)}{2\pi}\approx 5.5\times10^9\ \mathrm{m},\qquad r_2=\frac{(20{,}000)(864{,}000)}{2\pi}\approx 2.8\times10^9\ \mathrm{m}

The mass ratio is $$m_1/m_2=v_2/v_1=20/40=1/2$$. Star 1 is therefore half as massive as star 2.