Astronomy and Cosmology

Luminosity and radiant flux intensity: the inverse square law

The luminosity L of a star is the total power of radiation it emits: the energy it radiates in every direction each second, measured in watts. Luminosity belongs to the star alone and does not change with the observer's position. The Sun radiates 3.8 x 10^26 W, and the most luminous stars radiate more than a million times as much power as that. When a question says a star is 'brighter', it usually means something different, so always check whether it is the luminosity (a property of the star) or the flux intensity received (which depends on distance) that is being described.

The radiant flux intensity F is the power received per unit area on a surface placed at right angles to the radiation, measured in W m-2. It is what a detector at that distance actually measures, and unlike luminosity it falls off with distance. Picture a sphere of radius d drawn around the star: every part of the radiation crosses that sphere, whose surface area is 4 pi d^2. The power L is therefore spread over an area 4 pi d^2, so F = L / (4 pi d^2). This is the inverse square law: F is inversely proportional to the square of the distance, so doubling the distance reduces the flux intensity to a quarter, and multiplying it by ten reduces it to one hundredth.

Rearrange the same equation for the calculation you will meet most often: L = 4 pi d^2 F gives the luminosity if you know the distance, and d = square root of (L / (4 pi F)) gives the distance if you know the luminosity. That second form is the whole idea behind a standard candle, so make sure you can rearrange it confidently.

Astronomical distances are enormous, so they are quoted in parsecs and light years. One parsec is the distance at which 1 astronomical unit subtends an angle of 1 arcsecond, equal to 3.1 x 10^16 m. One light year is the distance light travels in a year, 9.5 x 10^15 m. Then 1 kpc = 3.1 x 10^19 m and 1 Mpc = 3.1 x 10^22 m. Convert to metres before you substitute into any of these equations.

• Luminosity L: total power of radiation emitted by the star, in W. A property of the star.
• Radiant flux intensity F: power received per unit area at right angles to the radiation, in W m-2. Depends on the distance.
• F = L / (4 pi d^2), so L = 4 pi d^2 F and d = square root of (L / (4 pi F)).
• 1 pc = 3.1 x 10^16 m, 1 ly = 9.5 x 10^15 m, 1 Mpc = 3.1 x 10^22 m.

Standard candles: turning a known luminosity into a distance

An object whose luminosity is known is called a standard candle. If you know its luminosity L and you measure the flux intensity F you receive from it, then d = square root of (L / (4 pi F)) hands you the distance. Measuring distances to galaxies therefore becomes a hunt for objects of known luminosity inside them.

The best known standard candles are Cepheid variables, yellow supergiants whose brightness rises and falls with a very regular period of a few days to about a hundred days. Henrietta Leavitt found in 1912, from Cepheids in the Small Magellanic Cloud, that the longer the period the greater the luminosity: the period-luminosity relation. A Cepheid is ideal because the period is simple to measure and, unlike its brightness, it does not change with distance. Measure the period, read the luminosity from the relation, measure the flux intensity, and the inverse square law gives the distance. Delta Cephei, the prototype, pulsates with a period of 5.37 days.

A worked example in the same style. A Cepheid has a luminosity of 4.8 x 10^29 W and its flux intensity at the Earth is measured as 2.3 x 10^-9 W m-2. Then d^2 = L / (4 pi F) = 4.8 x 10^29 / (4 pi x 2.3 x 10^-9) = 1.7 x 10^37 m2, so d = 4.1 x 10^18 m. Dividing by 9.5 x 10^15 m per light year gives 430 ly, or 131 pc. This is the kind of accuracy the method really delivers, and it is why Cepheids are the workhorse of nearby-galaxy distances.

Type Ia supernovae reach much further. A white dwarf in a close binary pulls matter off its companion until it reaches the Chandrasekhar limit of about 1.4 solar masses and explodes. Because the explosion always happens at nearly the same mass, the peak luminosity is nearly always the same, about 4 x 10^9 times the luminosity of the Sun, and the explosion is bright enough to be seen in a galaxy billions of light years away. Supernovae form the last rung of the distance ladder: parallax calibrates nearby stars, Cepheids calibrate the nearby galaxies, and those galaxies' supernovae calibrate the most distant ones.

Stellar radii: Wien's displacement law and the Stefan-Boltzmann law

The spectrum of a star is close to a black-body spectrum, and the wavelength at which it peaks tells you its surface temperature. Wien's displacement law states that the peak wavelength is inversely proportional to the surface temperature: lambda_max T = 2.9 x 10^-3 m K, where lambda_max is in metres and T is in kelvin. A hot star peaks in the blue, a cool star peaks in the red, and rearranging to T = (2.9 x 10^-3) / lambda_max turns a colour measurement into a temperature. Convert nanometres to metres first: a peak at 500 nm is 5.0 x 10^-7 m, giving 5800 K for the Sun.

The Stefan-Boltzmann law fixes how much power a given area of a star radiates: each square metre of surface radiates sigma T^4, with sigma = 5.67 x 10^-8 W m-2 K-4. A star of radius r has a surface area 4 pi r^2, so its total luminosity is L = 4 pi sigma r^2 T^4. Because the temperature is raised to the fourth power, a small change in temperature makes a large change in luminosity: doubling T makes a surface of the same area radiate sixteen times as much power.

Rearrange the law to find the radius: r = square root of (L / (4 pi sigma T^4)). Combine it with Wien's law and you can find the radius of a star without ever resolving its disc: the peak wavelength gives T, then the luminosity and T give r. The table below uses real stars and real values, and each row is consistent with both laws.

Emission and absorption spectra: measuring the redshift

A hot, low-pressure gas emits a line emission spectrum: bright lines at wavelengths that are unique to the elements in the gas, and which can be measured in the laboratory to high precision. A hot, dense source viewed through cooler gas produces a line absorption spectrum: a continuous spectrum crossed by dark lines at exactly the same wavelengths. Either way, the pattern of wavelengths is a fingerprint of the elements present, and the laboratory values are known to better than one part in a million.

When the spectrum of a distant galaxy or quasar is recorded, the lines are found at wavelengths longer than the laboratory values: they have been shifted towards the red end of the spectrum. This is called the redshift. All the lines are shifted by the same fraction, not just moved at random, and the fractional shift is what carries the information. If the lines are shifted to shorter wavelengths instead, the object is blueshifted and is approaching: the Andromeda galaxy is one of the few galaxies that shows a blueshift, approaching the Milky Way at about 110 km s-1.

The size of the shift is given by delta_lambda / lambda = delta_f / f = v / c, where lambda is the wavelength measured in the laboratory, delta_lambda is the increase in that wavelength, f is the frequency, c is 3.00 x 10^8 m s-1 and v is the speed of the source relative to the observer along the line of sight. The three ratios are all equal, so a redshift can be expressed as a fraction of the wavelength, of the frequency or of the speed of light. The quantity delta_lambda / lambda is called the redshift z. Because the ratio is dimensionless, the wavelengths can be left in nanometres as long as both are.

Note where this formula comes from and where it stops. For motion through space at speeds much less than c, it is the Doppler effect for light, and the frequency shift delta_f / f is the fractional change in the frequency received. For the most distant galaxies the light is redshifted because the space it travels through stretches while it is travelling, and the simple formula is a low-speed approximation; Cambridge uses it for all the redshifts you are asked about.

Hubble's law and the expanding Universe

Comparing the redshifts of many galaxies with their distances, Edwin Hubble found in 1929 that the recession speed of a galaxy is proportional to its distance from us. Written as an equation this is Hubble's law, v = H0 d, where the constant of proportionality H0 is the Hubble constant. A graph of v against d is a straight line through the origin, and its gradient is H0. Modern measurements give H0 = 2.2 x 10^-18 s-1, which is the same thing as 68 km s-1 Mpc-1: a galaxy 1 Mpc further away recedes 68 km s-1 faster.

Watch the units, because this is where most marks are lost. If you use v in km s-1 and d in Mpc, then v = H0 d gives H0 in km s-1 Mpc-1, not in s-1. Divide a speed in m s-1 by a distance in metres to get H0 in s-1, which is the form every other equation needs. A galaxy at 160 Mpc whose recession speed is 1.1 x 10^4 km s-1 is 4.96 x 10^24 m away travelling at 1.1 x 10^7 m s-1, so H0 = 1.1 x 10^7 / 4.96 x 10^24 = 2.2 x 10^-18 s-1.

Why does the redshift tell us the Universe is expanding? Every distant galaxy is receding from us, and the further away it is the faster it recedes. If the galaxies were flying through space away from a particular place, that place would be the centre of the Universe and we would have to be close to it, which is improbable. There is a better explanation: the space between the galaxies is itself stretching, carrying the galaxies apart. Light travelling through that stretching space has its wavelength stretched with it, which is exactly the redshift we observe.

This also explains why the expansion has no centre. Take any galaxy as your vantage point and you would see exactly the same pattern, every other galaxy receding with a speed proportional to its distance. No galaxy is more central than any other, and there is no edge to the expansion and nothing outside it. Hubble's law is a statement about the whole of space, not about the Milky Way's special position.

The Big Bang theory and the age of the Universe

If the galaxies are moving apart now, they must have been closer together in the past. Run the film backwards and there was a time when all the matter and energy of the Universe was packed into an extremely hot, extremely dense state, and the expansion of the Universe began from there. This is the Big Bang theory. The Big Bang was not an explosion of matter into empty space from a particular point: it was the beginning of the expansion of space itself, everywhere at once, which is why there is nothing to point at and call the site of the explosion.

Hubble's law gives an estimate of how long ago that was. If the recession speed has been constant, then the time taken for a galaxy to travel the distance d at speed v = H0 d is t = d / v = 1 / H0. With H0 = 2.2 x 10^-18 s-1 this gives t = 4.5 x 10^17 s, which is about 4.5 x 10^17 / 3.15 x 10^7 = 1.4 x 10^10 years, or about 14 billion years. Every galaxy gives the same answer, which is the point: 1 / H0 is the same for all of them, so it is a property of the Universe.

Treat that number as an estimate rather than a measurement. The calculation assumes the recession speed has never changed, and it has: the expansion was decelerated for the first several billion years by the gravitational attraction of the matter in the Universe, and it now appears to be accelerating. The formula also ignores the time during which the Universe was dense enough for light to be trapped. A smaller value of H0 gives a larger age, so a 10 per cent uncertainty in H0 gives roughly a 10 per cent uncertainty in the age.

Three observations support the Big Bang theory. Hubble's law itself is the first: the expansion implies a hot, dense beginning. The second is the cosmic microwave background, microwave radiation at about 2.7 K filling the whole sky, which is the cooled remnant of the radiation that filled the early, hot Universe, and which is the same in every direction because it comes from everywhere at once. The third is the measured abundance of the light elements: the Universe is about three quarters hydrogen and one quarter helium by mass, matching what calculations of the first few minutes of a hot, dense Universe predict.

Working with the equations

Every calculation in this topic starts from one of six equations, and the marks are usually for choosing the right one, substituting correctly and giving the unit. Learn them as a set, with the quantity each one gives you. The inverse square law and its rearrangements link luminosity, flux intensity and distance. Wien's law gives a temperature from a colour. The Stefan-Boltzmann law gives a luminosity from a radius and a temperature, or a radius from a luminosity and a temperature. The redshift formula gives a recession speed from a wavelength shift. Hubble's law gives a distance from a recession speed, or a recession speed from a distance. The reciprocal of the Hubble constant gives an age.

Two habits will save marks. First, put everything into SI units before substituting: nanometres into metres, Mpc into metres, km s-1 into m s-1. Second, keep the powers of ten under control by writing each quantity in the form a x 10^n before you divide, because it is very easy to lose a factor of 10^15 between a parsec and a metre. Check your answer for physical sense: a recession speed must be less than c, a stellar radius must come out between roughly 10^7 m and 10^12 m, and an age of the Universe must come out around 10^17 s or 10^10 years.