2 Estimating the distance to a cluster When you see a bright star in the sky at night, you don't really know if you're looking at a truly luminous star at a great distance from us or a faint star which is relatively close. Because of the inverse square law of the intensity of light, we can relate the apparent magnitude, m, of a star (its brightness at a wavelength A as measured from the Earth) to its absolute magnitude, M (its brightness at wavelength A as measured from a standardised distance of exactly 10 parsecs), by the relation: myM=5logD -5 (1) where D is the distance to the star in parsecs. The left-hand side of the equation (mx M3) is known as the Distance Modulus. If we can find this value for a star, we can find its distance (neglecting the effects of interstellar dust and gas which dim and redden the light coming from other stars). When we have a large sample of stars which we know are all at the same distance, then the D in Equation (1) is a constant for all those stars. Apparent magnitude is the quantity we can measure from Earth; absolute magnitude is what we need. Main sequence stars of a given colour should have the same absolute magnitude on average, so by comparing the apparent magnitudes of main sequence stars in one cluster to the absolute magnitudes of those in a sample of stars of known distance, you can calculate the distance modulus of the cluster. Main sequence stars of the same colour in both your selected cluster and the solar neighbourhood should have the same absolute visual magnitude. In the lectures we discussed the colour ratios, such as by /bg. In astronomy papers it is common to recast this ratio in terms of apparent magnitudes, mp = B and my = V. This turns the colour ratio into the colour (B-V) which is also large for red stars. This is the notation we will use in this lab. By shifting the colour-magnitude diagrams (a variant of the H-R diagram that refers specifically to a scatter plot of magnitude versus colour) of the two samples of stars in magnitude (vertically) until the main sequence bands coincide with one another, while keeping the colour scales aligned, you can estimate the distance modulus mv M. Astronomers call this technique main sequence fitting. 3. Below are two colour-magnitude-diagrams for Messier 3, a globular cluster, and the Pleiades, an open cluster. 14 Messier 3 0 0.5 1.5 2 16 The Pleiades . .. . ..s 6 T 6 18 8 TTTTT 8 10 10 20 12 12 -0.5 0.5 1.5 22 1 2 2.5 0.5 B-V 1.5 B-V Determine the approximate distance (give BOTH the distance modulus and distance in parsec) to EACH of the clusters by using the properties of the Sun; Mv=+4.83 and (B-V)=0.62