# General Astronomy/The Distance Ladder

The Cosmic Distance ladder—each step is less accurate that the one before

• Radar is the most accurate method, but it works only on solid bodies in the solar system. The Sun and gaseous planets such as Jupiter do not reflect radio waves and hence cannot be ranged this way.
• Stellar parallax, or distance based on how a star's apparent location varies with the observer's location, is the original method used to determine stellar distances. This is the most accurate method for estimating distances to stars, but is limited to only the nearest stars. Parallax is smaller the farther away a star is, and can be detected for only several thousand stars as viewed from Earth. However, parallax is detectable for many more by spacecraft above the Earth's obscuring atmosphere.
• Cepheid variables, is a method of determining cosmic distances based on the "period luminosity" characteristics of certain stars. Cepheids are stars whose total brightness (luminosity) varies with specific periods. They vary in brightness from brightest to dimmest to brightest in certain time periods.

It turns out that the period is related to the star's absolute magnitude or luminosity. Luminosity is a measure of how bright the star truly is at all wavelengths.[By comparison, light bulbs come in various luminosities, such as 60 watts or 100 watts. A 60 watt light bulb will appear much brighter at 5 feet than at 50 feet, but it will still have a 60-watt luminosity. If you know what the luminosity is, and can measure the apparent brightness, you can determine the distance by a simple mathematical relation.] By observing the star's period, we know its luminosity (which is directly related to another quantity called "absolute magnitude". By then comparing the known brightness [absolute magnitude] to the apparent brightness [apparent magnitude], the distance can be found by applying the inverse square law [just as you could determine the distance to a 60-watt light bulb if you could measure how bright it appears at your distance]. The specific equation is known as the "distance modulus." This is most useful for finding distances to stars clusters of which the Cepheid is a part. (Other stars with a similar relationship are the RR Lyra variables.)

• Spectroscopic parallax and related techniques based on spectral classification and the HR diagram (below). That is, if we know the spectral type of a "normal" star, we know its luminosity (actual total energy output at all wavelengths). This is because stars follow a pattern. That is, all G-type "normal" (non-Giant) stars are like all other G-type stars. All B-type normal stars are like all other B-type normal stars. More specifically, all G2 normal stars (such as our Sun) have approximately the same luminosity. Thus, if we can identify a star as being a G2 from its spectrum, then we know its luminosity, because we know that it is essentially the same as the Sun's. If we know the luminosity or absolute magnitude, we can compare that to the observed brightness [apparent magnitude] to deduce mathematically the actual distance by the distance modulus.
• Main Sequence fitting is a process related to "spectroscopic parallax" that compares the Hertzsprung -Russell (HR) diagram of the stars in a star cluster, with a calibrated HR diagram to determine absolute magnitude. This again is compared to the apparent magnitude of the stars and the distance determined by the distance modulus relation.

The processes above are useful for determining distances within the Milky Way Galaxy. Several other techniques for determining distances beyond our Milky Way Galaxy:

• Cepheid variables. The technique can be used for a few nearby galaxies, but is limited because outside of a few nearby galaxies, individual stars cannot be seen.
• Supernova magnitudes. Type Ia supernovae tend to be caused by the same series of events and all tend to be the same absolute magnitude. Since they are so bright (sometimes exceeding the brilliance of an entire galaxy) they can be seen much farther than individual stars and distance can be obtained through the distance modulus by comparing the apparent magnitude to the known absolute magnitude.
• The Tully-Fisher relationship. Astronomers have noted that the mass of spiral galaxy is related to its rotation rate. Mass is related to the number of stars, and the number of stars is related to the absolute magnitude of the galaxy (the greater the mass, the more stars and the brighter the galaxy). Thus by measuring rotation rate of the galaxy, and estimation of its true brightness can be made, and distance follows through the distance modulus.
• Red-Shift. Galaxies show shifts in the position of spectral lines in their light, dependent on how far the galaxy is. The farther the galaxy is, the more its light (and the spectral lines contained in that light) is shifted toward the red end of the spectrum. Thus by simply measuring the amount of shift, astronomers obtain an idea of how quickly the galaxy is moving away from us; given an assumed cosmological model this then allows the distance to the galaxy to be calculated.