## How bright is a star? Introducing luminosity and its use for calculating distances

If we have a detector on earth of 1 m^2, one can measure something called the “solar constant” or “apparent brightness” which is called “b_sun” ‘b’ as in brightness, a measure of the flux of energy per square meter of detector on earth.

b_sun = 1361 W/m^2 that is the intensity of the sun’s radiation in the vicinity of earth, approximately 100 ordinary incandescent light bulbs of 100 Watts on one square meter. The sun radiates its heat to space, some of it falls on earth the rest is lost to space.

This is to give you an idea of how to measure brightness. Basically, you have to imagine that the detector, instead of being on earth, that it could be placed around the star and then measure the flux of photons that are going through the surface of the detector.

Luminosity is defined as a quantitative measure of the brightness of a star. It is an intrinsic property of the star, it is the power of a star, i.e., the amount of energy (light) that a star emits from its surface in the visible spectrum. There is also bolometric luminosity which I won’t talk about which also take into account radiation outside the visible spectrum.

Luminosity is usually expressed in watts and measured in terms of the luminosity of the sun (L_sun). For example, the sun’s luminosity is 400 trillion trillion watts (energy emitted in joules per second, see the calculation below). One of the closest stars to Earth, Alpha Centauri A, is about 1.3 times as luminous as the sun! It means that if Alpha centauri was our sun, that is if it was physically placed at the same distance from earth, we would receive much more light per square meters here on earth. But how can astronomers measure the luminosity of distant stars? Let’s start by the calculation for the sun.

Apparent Brightness = Luminosity (Energy radiated by a light-source) / Area of the sphere over which the energy is spread

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

The brightness of the light from an object (L constant) is inversely proportional to the square of the distance D from the object. Twice as far means one-fourth as bright.

$b(D) \propto \frac{1}{D^{2}}$

The sun’s luminosity is on earth (at a distance D_sun):

(1) $L = 4 \pi D_{sun}^{2} b_{sun} \approx 4 \times 10^{26} W$

That’s, in power of tens, 400 trillion (10^12) trillion (10^12) watts.

Now, in terms of the flux of energy passing through one square meter of a sphere surrounding the sun, the luminosity can also be expressed as:

(2) L = area of the surface of the sun x flux “F” (W/m^2), so:

$4 \pi D_{sun}^{2} b_{sun} = 4 \pi R_{sun}^{2} \times F$

(3) $F = b_{sun}\frac{D_{sun}^{2}}{R_{sun}^{2}} = 6.29 \times 10^{7} \frac{W}{m^{2}}$

This is the amount of radiated energy by the sun. Every square meter of the sun produces 63 Mega Watts of light, so approximately, 600 thousand 100 watts light bulbs! Given that the sun is huge, that’s a lot of light/energy lost in space.

To measure the luminosity of DISTANT stars (remember that in the previous post, we saw that geometric parallax is only accurate for NEARBY stars), one has to measure the size of the star (or radius) and the star’s temperature. But in most cases neither can be measured directly. To determine a star’s radius, two other metrics are needed: the star’s angular diameter and its distance from Earth, often calculated using parallax. Both can be measured with great accuracy in certain special cases.

So we have 2 categories: stars with luminosity calculated from their geometric parallax and stars with luminosity estimated from their spectrum.

Cat 1: Brightness + distance + inverse square law for dimming allows to calculate intrinsic luminosity (see equation 1)

Cat 2: Scientists use spectroscopic parallax to measure the distance to stars, by assuming that spectra from distant stars of a given type are the same as those from nearby stars of the same type. They use the Hertzsprung-Russell diagram, which gives a place to each star according to the point it has reached in its lifecycle. This method enables scientists to estimate the luminosity of a star that is far away by comparing its spectrum to those of nearer stars.

NOTE: It should first be noted that spectroscopic parallax has nothing to do with geometric parallax. The use of the word parallax here is simply a reference to the goal of finding distances.

However for most stars the angular diameter or parallax, or both, cannot be measure with any certainty. Since the effective temperature is merely a number that represents the temperature of a black body that would reproduce the luminosity, it obviously cannot be measured directly, but it can be estimated from the spectrum.

1. Temperature: A black body radiates power at a rate related to its temperature, that is, the hotter the black body, the greater its power output per unit surface area. An incandescent or filament light bulb is an everyday example. As it gets hotter it gets brighter and emits more energy from its surface. The relationship between power and temperature is not a simple linear one though. The power radiated by a black body per unit surface area is varies with the fourth power of the black body’s effective temperature. The power output, for a perfect black body where sigma is a constant called the Stefan-Boltzmann constant. As a star is not a perfect black body we can approximate this relationship as:

$L \approx \sigma T^{4}$

This relationship helps account for the huge range of stellar luminosities. A small increase in effective temperature can significantly increase the energy emitted per second from each square metre of a star’s surface.

2. Size (radius): If two stars have the same effective temperature but one is larger than the other it has more surface area. The power output per unit surface area is fixed by equation so the star with greater surface area must be intrinsically more luminous than the smaller one.

Assuming stars are spherical then surface area S,where R is the radius of the star, is given by:

$S = 4 \pi R^{2}$

To calculate the total luminosity of a star we can combine these 2 equations:

$L \approx S \sigma T^{4}$

In practice this equation is not used to determine the luminosity of most stars as only a few hundred stars have had their R directly measured. If however, the luminosity of a star can be measured or inferred from other means (eg by spectroscopic comparison) then one can use that equation to determine the radius R of the star.

Let’s get back to distance calulations for Cat 2 stars. Given the equation above to calculate the luminosity of a given star, knowing the sun’s luminosity, the distance to the chosen star (D ‘star’) is given by solving this equation and with D_sun equal to 1 AU:

$\frac{L_{star}}{L_{sun}} = \frac{D_{sun}^{2}}{D_{star}^{2}} \times \frac{b_{star}}{b_{sun}}$

Spectroscopic parallax is only accurate enough to measure stellar distances of up to about 10 kpc. This is because a star has to be sufficiently bright to be able to measure the spectrum, which can be obscured by matter between the star and the observer. Even once the spectrum is measured and the star is classified according to its spectral type there can still be uncertainty in determining its luminosity, and this uncertainty increases as the stellar distance increases. This is because one spectral type can correspond to different types of stars and these will have different luminosities.

Finally, a special type of star called a Cepheid star is very important for measuring distances. With its 41.4-day pulsation period (its luminosity varies in a very regular way), RS Pup is one of the brightest known long-period Cepheids in the Galaxy. But this star also stands out as the only known example of the intimate association of a Cepheid with a dust cloud or reflection nebulae. In the following time-lapse video:

RS Puppis and a reflection nebulae

Observations from the NASA/ESA Hubble Space Telescope over a number of weeks show the variable star RS Puppis and its environment (a large cloud of gas). A phenomenon known as a light echo can be seen around the star, creating the illusion of gas clouds expanding out from RS Puppis. The cloud is not expanding, some of the pulsating light of the star RS Puppis is reflected by the surrounding gas cloud as seen in this video.

The light that travelled from the star to a dust grain and then to the telescope arrives a bit later than the light that comes directly from the star to the telescope. As a consequence, if one measures the brightness of a particular, isolated dust blob in the nebula, one will obtain a brightness curve that has the same shape as the variation of the Cepheid, but shifted in time.

By monitoring the evolution of the brightness of the blobs in the nebula, the astronomers can derive the blobs distance from the star: it is simply the measured delay in time, multiplied by the velocity of light (300 000 km/s). Knowing this distance and the apparent separation on the sky between the star and the blob, one can compute the distance of RS Pup from earth with great accurary. The distance of RS Pup was found to be 6500 light years, plus or minus 90 light years.

To conclude, one last type of star that can be used to measure distances in the universe is a Type 1a supernova. A supernova is an explosion that marks the end of life of certain stars with specific properties. A Type 1a supernova appears in a binary star system in which a white dwarf gather matter from a normal companion star via accretion. Matter streaming from the companion star accumulates on the white dwarf until the white dwarf explodes. While one star is going supernova, the other normal star possibly continues to exist.

Both Cepheid and Type 1a supernova are called “standard candles” because for the former, the pulsation is regular no matter where cepheids are located. For the later, luminosity is assumed to be the same for all Type 1a supernovae, no matter where they are located in the universe. (ref)

APPENDIX:

• A star can be approximated to a black body object (an object which absorbs all light that reaches it). The Stefan-Boltzmann equation gives the relationship between radiated energy from a black body as it heats and its temperature. Flux of radiated energy from the black body = sigma constant times temperature^4. Knowning the flux of the sun, one can calculate its surface temperature, that is 5770K for the photosphere.