Stars: radiation, distances and the HR diagram flashcards

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Black body
An idealised object that absorbs all the electromagnetic radiation falling on it; its emitted spectrum depends only on its temperature.
Luminosity L
The total power output of a star: the energy it radiates per second in all directions (W).
Intensity I
Power per unit area incident at right angles to the direction of the radiation (W m⁻²).
Parallax angle p
Half the angular shift of a nearby star against distant stars over six months: the angle at the star subtended by 1 AU.
Parsec (pc)
The distance at which 1 AU subtends an angle of 1 arcsecond. 1 pc = 3.09 × 10¹⁶ m ≈ 3.26 ly.
Light-year (ly)
The distance light travels in a vacuum in one year: 9.46 × 10¹⁵ m.
Standard candle
An astronomical object of known luminosity, so its distance can be found from the intensity received at Earth.
Hertzsprung–Russell (HR) diagram
A plot of luminosity (log scale) against surface temperature (log scale, decreasing to the right) for stars.
Wien's displacement law
λmax T = 2.898 × 10⁻³ m K (T in kelvin). (Yes)
Stefan–Boltzmann law
L = σAT⁴, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴. (Yes)
Luminosity of a spherical star of radius r
L = 4πr²σT⁴ (A = 4πr²). (No — learn it)
Intensity at distance d from a star
I = L/4πd² (Yes)
Distance by parallax
d = r/tan p ≈ r/p (r = 1 AU, p in radians); in parsecs d = 1/p with p in arcseconds. (No — learn it)
Distance from a standard candle
d = √(L/4πI) (No — derive it)
Black-body curve: effect of a higher temperature
Curve higher at every wavelength, peak at a shorter wavelength, area (∝ T⁴) much larger.
What does the area under a black-body curve represent?
The total power emitted per unit area of the surface, proportional to T⁴.
HR diagram axes
Vertical: luminosity (or L/L☉), log scale. Horizontal: surface temperature, log scale, hottest on the LEFT.
Positions on the HR diagram
Main sequence: diagonal band. Red giants: above it, right. Supergiants: across the top. White dwarfs: bottom left.
Lines of constant radius on an HR diagram
Diagonal lines (L ∝ T⁴ at fixed r); stars get larger towards the top right.
Path of a Sun-like star on the HR diagram
Main sequence → up and right to red giant → (planetary nebula) → white dwarf at bottom left, fading down and right.
Why is a red giant luminous although it is cool?
L = 4πr²σT⁴: its enormous surface area more than makes up for the low temperature.
Why is a white dwarf dim although it is hot?
Its radius is tiny (about Earth-sized), so its surface area is very small.
Why does parallax work only for nearby stars?
The angle is ∝ 1/d; for distant stars it is too small to measure against its uncertainty.
Why are parallax measurements taken six months apart?
The Earth is on opposite sides of its orbit: the largest baseline (2 AU) and the largest shift.
What decides how a star's life ends?
Its mass. Up to about 8 M☉: white dwarf. More massive: supernova, then neutron star (core ≈ 1.4–3 M☉) or black hole (core > ≈ 3 M☉).
Why do massive stars have shorter lives?
They are far more luminous, so they use up their core hydrogen much faster despite having more of it.
Luminosity vs apparent brightness
Luminosity is the star's power output; apparent brightness (intensity at Earth) also depends on distance, I = L/4πd².
A star's spectrum peaks at 480 nm. Surface temperature?
T = 2.898 × 10⁻³ ÷ 480 × 10⁻⁹ = 6040 K.
Luminosity of a star of radius 7.0 × 10⁸ m and surface temperature 6000 K?
L = 4π(7.0 × 10⁸)² × 5.67 × 10⁻⁸ × 6000⁴ = 4.5 × 10²⁶ W.
Intensity at Earth from a star of luminosity 4.0 × 10²⁷ W at 5.0 × 10¹⁷ m?
I = 4.0 × 10²⁷ ÷ (4π × (5.0 × 10¹⁷)²) = 1.3 × 10⁻⁹ W m⁻².
Parallax angle 0.40 arcseconds. Distance in pc and in m?
d = 1 ÷ 0.40 = 2.5 pc = 2.5 × 3.09 × 10¹⁶ = 7.7 × 10¹⁶ m.
Star X: radius 2.0 times and temperature 1.5 times those of star Y. Ratio L_X/L_Y?
L ∝ r²T⁴: 2.0² × 1.5⁴ = 4 × 5.06 = 20.
Standard candle of luminosity 3.0 × 10²⁹ W, measured intensity 6.0 × 10⁻¹² W m⁻². Distance?
d = √(3.0 × 10²⁹ ÷ (4π × 6.0 × 10⁻¹²)) = 6.3 × 10¹⁹ m.
Before using Wien or Stefan–Boltzmann…
Temperature in kelvin (°C + 273) and wavelength in metres (nm × 10⁻⁹).
Finding d from I = L/4πd²
Rearrange in symbols first: d = √(L/4πI). Don't forget the square root or the square on d.
Percentage uncertainty in L = 4πr²σT⁴
%L = 2 × %r + 4 × %T: the fourth power makes temperature errors dominate.
Sketching an HR diagram
Label both axes with quantities, hottest temperature on the left, and mark main sequence, red giants, supergiants and white dwarfs.

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