23.3 Standard Candles as Distance Indicators

Standard Candles

A standard candle is an astronomical object whose absolute luminosity (intrinsic brightness) is known. Because we know how bright the object truly is, we can compare this to how bright it appears from Earth and calculate its distance using the inverse square law:

$b=\frac{L}{4\pi d^2}$

where:

• $b$ = apparent brightness (W m⁻²)
• $L$ = absolute luminosity (W)
• $d$ = distance to the object (m)

Rearranging to find distance:

$d=\sqrt{\frac{L}{4\pi b}}$

Examples of Standard Candles

Cepheid Variables

Cepheid variables are pulsating stars whose period of pulsation is directly related to their average luminosity (the Period–Luminosity relationship, discovered by Henrietta Leavitt). By measuring the period, astronomers determine $L$; by measuring $b$, they calculate $d$. Edwin Hubble used Cepheids in the Andromeda galaxy to prove it was a separate galaxy far beyond the Milky Way.

Type Ia Supernovae

A Type Ia supernova occurs when a white dwarf accretes mass from a companion star until it reaches the Chandrasekhar limit (1.4 solar masses) and explodes. Because the triggering mass is always the same, all Type Ia supernovae reach nearly the same peak luminosity ($10^{43}$ W). This makes them visible across billions of light-years, far beyond the reach of individual Cepheids.

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Blackbody Radiation and Wien's Displacement Law

Stars behave approximately as black bodies — objects that absorb all incident radiation and emit a characteristic spectrum that depends only on temperature.

Wien's Displacement Law

The wavelength ${\lambda }_{max}$ at which a black body emits maximum power is inversely proportional to its absolute temperature $T$:

${\lambda }_{max}T=b_W\approx 2.9\times 10^{-3}\text{ m K}$

where $b_W$ is Wien's displacement constant.

Application: By measuring the peak wavelength of a star's spectrum, we can determine its surface temperature:

$T=\frac{2.9\times 10^{-3}}{{\lambda }_{max}}$

Example: The Sun's peak emission is at ${\lambda }_{max}\approx 500$ nm: $T_{\odot }=\frac{2.9\times 10^{-3}}{500\times 10^{-9}}\approx 5800\text{ K}$

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Stefan-Boltzmann Law

The total power radiated per unit surface area of a black body is proportional to the fourth power of its absolute temperature:

$E=\sigma T^4$

where:

• $E$ = power radiated per unit area (W m⁻²)
• $\sigma =5.67\times 10^{-8}$ W m⁻² K⁻⁴ (Stefan-Boltzmann constant)
• $T$ = absolute temperature (K)

For a star of radius $R$, the total luminosity is:

$L=4\pi R^2\sigma T^4$

Example: If the temperature of a star doubles, its luminosity increases by a factor of $2^4=16$.

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Estimating the Radius of a Star

By combining Wien's displacement law (to find $T$) and the Stefan-Boltzmann law (relating $L$, $R$, and $T$), we can estimate a star's radius:

Step 1: Measure ${\lambda }_{max}$ from the star's spectrum → find $T$ using Wien's law.

Step 2: Measure apparent brightness $b$ and distance $d$ → find luminosity $L=4\pi d^{2}b$.

Step 3: Rearrange the Stefan-Boltzmann luminosity formula:

$R=\sqrt{\frac{L}{4\pi \sigma T^4}}$

Worked Example

A star has ${\lambda }_{max}=290$ nm and luminosity $L=3.9\times 10^{26}$ W. Estimate its radius.

Step 1: $T=\frac{2.9\times 10^{-3}}{290\times 10^{-9}}=10,000$ K

Step 2: $R=\sqrt{\frac{3.9\times 10^{26}}{4\pi \times 5.67\times 10^{-8}\times (10^4{)}^4}}$

$R=\sqrt{\frac{3.9\times 10^{26}}{4\pi \times 5.67\times 10^{-8}\times 10^{16}}}\approx \sqrt{\frac{3.9\times 10^{26}}{7.12\times 10^9}}\approx \sqrt{5.48\times 10^{16}}\approx 2.3\times 10^8\text{ m}$