Stellar Fingerprints

When you look at the night sky from your backyard, do you sometimes think that there is no order to all of those stars out there? If the star isn’t part of a well known constellation, is it nothing more than a point of light in a sea of other points of light? Nothing that distinguishes it from any other star?

Well, this just isn’t so. In fact, stars have characteristics such as temperature, luminosity (brightness), mass, galactic location, distance to the earth, and even age — all combined forming a stellar fingerprint that uniquely identifies a specific star.

You probably already know this, but did you ever stop to wonder how we came to know these unique characteristics of a star? After all, we can’t run up and stick a thermometer in a star, or run a tape measure from the star to Earth. So, how do we find get information about stars?

Finding the distance to a star

Well, this one had me the most curious, so this is the one I’ll take first. How do we measure the distance to a specific star? If the stars are nearby, we use stellar parallax

When you move towards objects that are near you, they seem to move in relation to the objects that are located much futher than you. You might notice this when you look at signs by the side of the road in comparison to the background detail when you’re traveling in a car. You can also notice this effect when you hold a pencil in front of you and view it through one opened eye and then another (see diagram).

This same effect seems to happen to stars that are close to the Earth. If you measure the angle to a star from a fixed point on the Earth, and then measure it again from the same point when the Earth is at the opposite position in its orbit around the sun (in 6 months time), you’ll find that the two measurements form a triangle where they intersect (see U of Oregon Diagram). If you half the triangle and then take the angle of one half, you’ll get a value in arcseconds (an arcsecond is 1/360 of a degree). You can then find the distance to the star using stellar parallex:

d = 1/p

The distance to the star (in parsecs, roughly equal to 3.26 light years) is equal to the inverse of the parallex angle of the star.

A light year is the distance light travels within a year — roughly 300,000 km/s

Using this approach we’ve been able to find the distances to several stars such as Proxima Centauri at 0.772 parallax (4.22 light years); Sirius B at 0.379 parallax (8.61 light years); and Epsilon Indi at 0.276 parallax (11.82 light years).

Of course, this approach works only for stars that are relatively close to the solar system, but once you have this information, you can use the distance in other calculations — such as to find the luminosity of a star.

Finding Luminosity

A star’s brightness is a measure of its luminosity.

Luminosity is the amount of light energy emitted by the star within a second, measured in watts (joules per second).

You might think that luminosity is directly related to the distance of the object from the Earth. Well, it is, but there are other factors involved such as the mass of the star and its temperature. If star A is further from the Earth than star B, but star A is much, much brighter, it can appear more bright to us than the closer star.

Still, the distance to the star can tell us its luminosity, with a simple formula:

L = 4pid2b
In this, the Luminosity is equal to the distance squared, multiplied by the brightness, and then multiplied by 4 times pi (pi approx equal to 3.1415926...). The brightness is the apparent brightness as its measured here on Earth (or wherever the viewpoint is), through techniques such as photometry. The brightness of a star is usually described by comparing it to Sirius A, the brightest star we see from Earth (and with a brightness of 1.0).

A simplified approach to finding luminosity is to plug the Sun's brightness, distance, and luminosity into the formula and then take the ratio of the two equations. By doing this, the value of 4pi falls out of the formula:

L/Lsun = (d/dsun)2 b / bsun

Luminosity can now be found by direct comparison between the star and the Sun.

For instance, if a star has a brightness of 5.2 x 10-12 compared to the sun, and it’s distance from earth is 5.2 x 106 that of the Sun to the Earth, you would use the following to find the luminosity:

Lstar/Lsun = (5.2 x 106)2 5.2 x 10-12 = 140

The star (Regulus) has 140 times the luminosity of the Sun, but appears dimmer because of its distance. You could use this same approach with any two stars — find the ratio of the stars and then solve for the unknown value:

L1/L2 = (d1/d2)2 b1/b2

With this, if you find out that star 1 is 3 times the distance of star 2 and appears twice as bright, you can figure the luminosity without having to use a calculator: star 1 has 18 times the luminosity as star 2.

Another characteristic you can find out about a star from the light it emits is its temperature, found next.

Finding a star’s temperature

Quiz time: which is hotter, a blue star or a red star?

The answer might surprise you — the blue star is hotter. The blue color is because most of the star’s radiation is in shorter wavelengths, hence in the blue to ultraviolet range. A cooler star has a longer wavelength, in the red to infrared range.

Wien’s Law states that as a star’s temperature increases, it’s color shifts to the blue.

You can find the temperature of a star by finding the wavelength of its maximum intensity, and using this value in the Wien’s Law equation:

wavelengthmax = .0029 / T

In the equation just shown, the maximum wavelength emission is equal to a constant value (.0029) divided by the temperature. The maximum wavelength emission can be found using instruments on Earth, so this value is used to find the star’s temperature:

T = .0029 / wavelength max

If a star has a maximum wavelength of 500 nm (5 x 10-7 m), its temperature would then be about 5800 degrees kelvin:

T = 0.0029 / 5 x 10-7

This is the temperature of our own Sun. Its color is due to the fact that the maximum wavelength emission is at 500nm, putting it within the yellow color range in the visible light spectrum.

You can find the maximum wavelength emission of any star using photometry, regardless of its distance from the Earth.

Of course, once you have a star’s temperature, and its luminosity, you can then find its radius.

Finding a star’s radius

Okay, let’s recap what we’ve been able to find out about distant stars.

We’ve been able to find their distance (if close enough to use stellar parallax), as well as their luminosity (regardless of distance). We can also find a star’s maximum wavelength emission, and we’ve used this to find the star’s color as well as temperature. One thing we haven’t found, yet, is a star’s size. We have found, though, the values necessary to find the radius of the star: its luminosity and it’s temperature.

A star’s luminosity is equal to its radius, squared, multiplied by its temperature to an exponent of 4:

L = 4piR2(const)T4

The (const) value in the equation is the Stefan-Boltzmann constant, a value of 5.67 x 10-8 W m-2 K-4. (Find other constants.)

You don’t have to remember this rather computationally instensive formula if you look at it as a measure of the ratio between the star and the Sun:

L/Lsun = (R / Rsun)2 (T / Tsun)4

Re-arranging this to search for the radius, you have:

R/Rsun = (Tsun / T)2 SQRT(L / Lsun)

For instance, the star Rigel has a temperature 3 times that of the Sun, and a luminosity 64,000 times that of the Sun (one very bright star). It’s radius in comparison to the Sun’s is:

RRigel/Rsun = (1/3)2SQRT(64,000) = 27.5

Rigel has a radius about 28 times that of our Sun. As the Sun’s radius is 6.96 x 105 km, Rigel’s radius would be about 1.9 x 107 km.

An so on…

There are other things we can find out about stars, but this should give you an idea of what we know, and what we can find out about a specific star.

And we didn’t even have to leave our backyards to find it.

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