understand the use of standard candles to determine distances to galaxies

Standard Candles – A‑Level Physics (9702)

Learning objective

Explain how astronomers use standard candles to determine the distances to galaxies and show how these methods fit into the full cosmic distance ladder.

1. The cosmic distance ladder

Each rung of the ladder provides a technique that is calibrated by the rung below it. The ladder therefore builds a self‑consistent distance scale from the Solar System to the edge of the observable Universe.

Rung	Typical distance range	Method (example)	What it measures
Parallax	≤ 0.1 kpc	Trigonometric parallax (Gaia, Hipparcos)	Geometric distance – zero‑point of the ladder
Standard candles	0.1 kpc – several Gpc	RR Lyrae → Cepheids → Type Ia supernovae	Known absolute magnitude → distance via the distance modulus
Secondary (empirical) indicators	≈10 – 200 Mpc	Tully‑Fisher, Surface‑Brightness Fluctuations, Fundamental Plane	Correlations between observable galaxy properties and luminosity
Hubble flow	≥ 30 Mpc (≈ z < 0.1)	Red‑shift + Hubble’s law	Recession velocity gives distance once H0 is known

Standard candles bridge the gap between the nearby parallax scale and the far‑away Hubble‑law regime.

2. What is a standard candle?

A standard candle is an astronomical object whose absolute magnitude M (or intrinsic luminosity) is known independently of its distance. By comparing M with the observed apparent magnitude m we obtain the distance.

3. The distance modulus

For a distance d expressed in parsecs (pc):

\[ m - M = 5\log_{10}(d) - 5 . \]

Re‑arranged:

\[ d = 10^{\frac{m-M+5}{5}}\;\text{pc}. \]

If d is given in megaparsecs (Mpc):

\[ \mu \equiv m-M = 5\log_{10}(d_{\text{Mpc}}) + 25 . \]

4. Main standard candles used for galaxies

Standard candle	Typical absolute magnitude M (band)	Uncertainty in M	Key property	Usable distance range
RR Lyrae	+0.6 (V)	±0.10 mag	Horizontal‑branch stars; almost constant M	0.001 – 0.1 Mpc
Cepheid variables	−3 to −6 (V)	±0.15 mag (after metallicity correction)	Period–luminosity (P–L) relation	0.01 – 30 Mpc
Tip of the Red Giant Branch (TRGB)	−4.0 (I)	±0.10 mag	Sharp cut‑off in the red‑giant luminosity function	0.1 – 20 Mpc
Type Ia supernovae	−19.3 (B)	±0.12 mag (after stretch‑colour correction)	Standardised peak luminosity of thermonuclear explosions	10 – 3000 Mpc

5. Cepheid variables – the classic standard candle

Cepheids pulsate with a period P (days) that is tightly correlated with absolute magnitude. A widely used V‑band relation is

\[ M_V = -2.81\log_{10}(P) - 1.43 \;\pm\;0.15\;\text{mag}, \]

where the quoted uncertainty includes the metallicity correction (metal‑rich Cepheids are ≈0.1 mag brighter).

Steps to obtain a distance

1. Identify the variable star and obtain a well‑sampled light curve.
2. Measure the pulsation period P from the light curve.
3. Use the P–L relation (plus any metallicity correction) to find M_V.
4. Measure the apparent magnitude m_V and correct for interstellar extinction: \[ m_V^{\text{corr}} = m_V - A_V . \]
5. Insert m_V^{\text{corr}} and M_V into the distance‑modulus to obtain d.
6. Propagate the uncertainties from the period, extinction, and zero‑point to give an error on d.

Example – Cepheid in a nearby galaxy

Given: P = 10 days, extinction‑corrected apparent magnitude m_V = 24.0 mag.

• Absolute magnitude: \[ M_V = -2.81\log_{10}(10) - 1.43 = -4.24\;\text{mag}. \]
• Distance modulus: \[ \mu = m_V - M_V = 24.0 - (-4.24) = 28.24 . \]
• Distance: \[ d = 10^{\frac{28.24+5}{5}}\text{ pc}=10^{6.648}\text{ pc}\approx4.5\times10^{6}\text{ pc}=4.5\text{ Mpc}. \]
• Uncertainty: ±0.15 mag in M_V → roughly ±7 % in distance.

6. Type Ia supernovae – extending the ladder

Type Ia supernovae result from the thermonuclear explosion of a carbon‑oxygen white dwarf that approaches the Chandrasekhar limit (≈1.4 M☉). Their peak absolute magnitude is uniform after two empirical corrections:

• Stretch (light‑curve shape) – broader light curves are intrinsically brighter.
• Colour (reddening) – removes the effect of dust extinction.

After these corrections the calibrated absolute magnitude is

\[ M_B = -19.30 \pm 0.12\;\text{mag}. \]

Using the distance modulus gives distances out to several gigaparsecs, enabling measurements of the Hubble constant and the acceleration of the Universe.

7. Secondary (empirical) distance indicators

These methods are calibrated by standard candles and are useful when individual standard candles cannot be resolved.

Indicator	Observable quantity	Physical basis	Typical distance range
Tully‑Fisher relation	Rotational velocity from 21 cm line width	Luminosity ∝ (rotation speed)4 for spiral galaxies	10 – 200 Mpc
Surface‑Brightness Fluctuations (SBF)	Pixel‑to‑pixel variance in an unresolved stellar population	Fluctuation amplitude falls off as 1/distance	≈ 20 – 100 Mpc (early‑type galaxies)
Fundamental Plane	Combination of effective radius, surface brightness, and velocity dispersion	Virial theorem applied to elliptical galaxies	≈ 50 – 200 Mpc

8. Redshift, recession velocity and Hubble’s law

For a galaxy whose spectral lines are shifted by a factor z, the observed wavelength λ_obs = (1+z)λ_rest. At low redshift (z ≲ 0.1) the Doppler approximation gives

\[ v \approx cz , \]

where c = 3.00 × 10⁵ km s⁻¹. In this regime the recession velocity is proportional to distance:

\[ v = H_0 d . \]

Key points for the syllabus:

• The linear form of Hubble’s law holds only while the expansion is “local” (≈ z < 0.1). At larger redshifts the relation must include the cosmological model (Ω_m, Ω_Λ) and the full relativistic expression for the luminosity distance.
• Combining a standard‑candle distance (from the distance modulus) with the measured redshift yields a value of H₀. Conversely, once H₀ is known, the redshift alone can give a distance for very distant galaxies.
• Redshift also provides a “look‑back time” – the time elapsed between the emission of the light and its detection. For small z, the look‑back time is ≈ z × (1/H₀), giving an age of the Universe of order 13–14 Gyr.

9. Sources of uncertainty and systematic error

• Interstellar extinction – dust absorbs and reddens light; corrections use colour excess or infrared observations.
• Metallicity effects – the Cepheid P–L zero‑point shifts by ≈0.1 mag between metal‑poor and metal‑rich environments.
• Calibration errors – the absolute zero‑point of each indicator is anchored to geometric distances (e.g., Gaia parallaxes for Cepheids, detached eclipsing binaries for the LMC).
• Malmquist bias – at large distances only the intrinsically brightest objects are detected, biasing distance estimates low.
• Statistical uncertainties – propagated from measurement errors in m, period, line width, colour, and from the intrinsic scatter of each relation.

10. Summary

• The distance modulus converts an observed apparent magnitude into a distance once the absolute magnitude is known.
• Cepheid variables and Type Ia supernovae are the principal standard candles for extragalactic work, covering complementary distance ranges from a few kiloparsecs to several gigaparsecs.
• Secondary indicators (Tully‑Fisher, SBF, Fundamental Plane) extend the ladder where individual candles cannot be resolved.
• At the largest scales the Hubble flow (redshift = cz) provides distances, and the combination of standard‑candle distances with redshifts yields the Hubble constant.
• Accurate distances require careful correction for extinction, metallicity, and calibration, and an awareness of systematic biases such as Malmquist bias.