Tag: parallax

How Far the Sun

How do we know our Sun is 93 million miles (150 million km) away1?

The ancient Greek astronomer and mathematician Aristarchus of Samos, who lived around 2,300 years ago, was probably the first person who made a reasonable attempt to determine the distance to the Sun.

Using a method of geometric analysis developed by Euclid (trigonometry had not yet been invented), Aristarchus measured the angle between the half-lit Moon and the Sun and determined that the Sun is 18 to 20 times farther away than the Moon.  Though he fell far short of the actual value of 389 due to the extreme difficulty of making accurate measurements using the instruments and methods available to him, Aristarchus showed the way for future generations of astronomers to determine the true distance to the Sun.

Determining the actual distance (and not the relative distance) to the Sun had to wait for Kepler’s Third Law of planetary motion that relates a planet’s orbital period to its distance from the Sun, the invention of the telescope, and Isaac Newton’s laws of motion and gravitation.

P^{2}\propto a^{3}

Distances within the solar system can be determined using trigonometry and parallax, which is the apparent shift of an object against the distant background stars as seen from different locations.

Hold your thumb at arm’s length and alternate between right and left eye open to see the parallactic shift.
Bring your thumb closer, and the shift is greater.

Measuring the parallax to a Sun-orbiting object (such as Mars) from two different locations on the Earth’s surface allows us to measure its distance and, thanks to Kepler and Newton, sets the scale for the entire solar system.  The true distance of each planet from the Sun can then be mathematically determined.  This was first accomplished in 1672, and has been done many times since, with ever-improving accuracy.

Observations of the position of Mars by Giovanni Cassini at Paris and Jean Richer at Cayenne
allowed the first determination of the distance to Mars using trigonometric parallax in 1672.

Today, we have even better methods to determine the scale of the solar system: timing radar reflections off of solar system objects, and measuring travel time for radio communications between Earth and spacecraft.  Both radar and radio signals travel at the speed of light, which is very well determined.

1Approximate average distance

Spectroscopic Parallax

For the nearest stars, the change in the position of the Earth in its orbit results in a tiny shift in the position of the nearby star relative to the distant background stars. This shift is called the trigonometric parallax. You can see the effect by holding your thumb up at arms length, closing your left eye, and lining up your thumb with something across the room. Now, alternate back and forth between having your right eye open and your left eye open and you’ll see the position of your thumb shift relative to an object further away. Move your thumb closer, and the shift is larger. That is the essence of trigonometric parallax.

Trigonometric Parallax

The distance to the star in parsecs (1 pc = 3.26 ly) is just

Now, a star’s distance, apparent brightness, and “true” (or intrinsic) brightness are related in the following way:

M = m + 5 (1 – log d)

where M = the absolute magnitude of the star

and m = the apparent magnitude of the star

and d = the distance to the star in parsecs

The absolute magnitude is the apparent magnitude the star would have if it were at a distance of 10 parsecs. Looking at it another way, the absolute magnitude is a proxy for the intrinsic brightness. The apparent magnitude is the star’s apparent brightness (as seen from Earth).

While the above equation is highly useful for general purpose calculations, to get the most accurate values astronomers must take into account atmospheric and interstellar extinction. And, anytime we deal with a star’s luminosity and its apparent brightness at some distance, d , we must specify the photometric system and optical filter that is being used. Or, less commonly (for practical reasons), we specify that the star’s luminosity and apparent brightness is to include all wavelengths of the electromagnetic spectrum, thus bolometric magnitudes are to be used.

Spectroscopic parallax is a bit of a misnomer, but here’s how it works for approximating the distance to main-sequence stars that are too far away to exhibit a measurable, reasonably certain, trigonometric parallax: measure the apparent magnitude of the star, and then using its spectrum to find its position on the H-R diagram, read off its absolute magnitude. Using your measured apparent magnitude and the star’s estimated absolute magnitude, you can solve for d the distance in the above equation.

Hertzsprung–Russell (H-R) diagram

The star’s color (the x-axis on the H-R diagram) is easy to measure, but a deeper analysis of the spectral lines is needed to determine whether the star is a main-sequence, giant, or supergiant star (or something else).