Hot gases emit light producing line spectra

When a gas is heated to a high temperature, its electrons can move to higher energy levels. As electrons fall back down to lower energy levels, they release energy as photons. The particular energies of the photons depend on the quantised energy level transitions in that element.

If this light is passed through a diffraction grating, the result is a line emission spectrum - a series of sharp, coloured spectral lines against a dark background. Each coloured line corresponds to a particular wavelength of light.

• Emission lines are quantised and element-specific, allowing identification of gases
• Line positions can be used to calculate photon energies using diffraction grating equation

Absorption spectra from white light

A line absorption spectrum is produced when white light passes through a cool, low-pressure gas.

• At lower temperatures, most electrons occupy ground state
• Photons of specific wavelengths are absorbed, exciting electrons up to higher levels
• Missing wavelengths appear as dark lines against the continuous spectrum

Emission and absorption spectra are complementary - absorption lines align with emission lines. This allows identification of elements.

Stellar absorption spectra occur due to starlight passing through outer gas layers. Comparing absorption lines with laboratory spectra enables stellar composition analysis.

Relating luminosity, radius and temperature

A star's luminosity (L) is its total power output. Stefan's Law gives:

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

Where:

• L = Luminosity (W)
• R = Radius (m)
• $\sigma$ = Stefan-Boltzmann constant (5.67 x 10^-8 W m^-2 K^-4)
• T = Surface temperature (K)

Wien's law

Wien's displacement law describes the relationship between the temperature of a blackbody and the wavelength at which it emits radiation most intensely.

The relationship between temperature and the max wavelength is given below:

$\lambda_{max}\text{ T = constant = 2.9 x 10}^{-3}\text{ m K}$

Where:

• $\lambda_{max}$ = peak wavelength (m)
• T = absolute temperature of blackbody(K)
• 2.9 x 10^-3^ m K = Wien's displacement constant

Worked example - Estimating the radius of Betelgeuse

Betelgeuse has a luminosity of 1.2 x 10³¹ W and a peak wavelength of 970 nm. Calculate the radius of Betelgeuse.

Step 1: Convert nm to m

to convert from. nm to m, multiply by 1 x 10^-9

970 nm = 970 x 10^-9 m

Step 2: Wien's law formula

$\lambda_{max}T = 2.9 \times 10^{-3}$

Step 3: Rearranged formula

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

Step 4: Substitution and correct evaluation

$T = \frac{2.9 \times 10^{-3}}{970 \times 10^{-9}} = 2989.7 \text{ K}$

Step 5: Stefan’s law formula

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

Step 6: Rearranged formula

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

Step 7: Substitution and correct evaluation

$R = \sqrt{\frac{1.2 \times 10^{31}}{4\pi \times 5.67 \times 10^{-8} \times (2989.7)^4}}\text{ = 4.6 x 10}^{11}\text{ m}$