Galaxies Hubbleʼs Law Author: Sarah Roberts
Hubbleʼs Law Introduction The first galaxies were identified in the 17th Century by the French astronomer Charles Messier, although at the time he did not know what they were. Messier, a keen observer of comets, spotted a number of other fuzzy objects in the sky which he knew were not comets. Worried that other comet hunters might be similarly confused, he compiled a list to prevent their misidentification. Messiersʼ list (where objects are identified by M for Messier, followed by a number, e.g. M51) contained information on 110 star clusters and spiral nebulae (galaxies) but it was almost 300 years before astronomers worked out what the fuzzy spiral nebulae actually were. Some people argued that these nebulae were island universes - objects like our Milky Way galaxy, but external to it. Others disagreed, and thought that these spiral objects were clouds of gas within the Milky Way, believing that ours was the only Galaxy in the Universe. The argument went on until the 1920s, when the American astronomer Edwin Hubble finally measured the distance to one of these spiral nebulae. Hubbleʼs Discovery In 1923 Hubble was studying novae (variable stars which have sudden, unpredictable outbursts) in M31 (now known as the Andromeda Galaxy), when he realised that one of the objects he was observing was in fact a Cepheid variable. Cepheids are stars whose brightness changes periodically over time, and they had been discovered by the American astronomer, Henrietta Leavitt, in the early 1900ʼs. Leavitt found what is now known as the Period-Luminosity (P-L) relationship, a link between the luminosity (brightness) of a Cepheid and its period. By measuring the period of a Cepheid (by observing itʼs brightness changes over several days or weeks), the P-L relationship can be used to determine itʼs actual brightness. Comparing this to the observed brightness, and assuming the difference is just caused by how far away the star is (see Inverse Square Law at end), you can estimate the distance. Hubble used the P-L relation to find the distance to the Cepheid he was studying in M31, and proved that it was located outside of our own Galaxy. This finally ended the debate on the nature of the spiral nebulae they were indeed distant galaxies like our Milky Way. Hubble continued his study of galaxy distances, using Cepheids as his measuring tool, before publishing his results in 1929. In his paper, Hubble plotted a graph of the velocity of galaxies (obtained from determining the redshift of the spectra of these galaxies) against their distances (from Cepheid variabes), as shown in Figure 1. This plot showed that most galaxies are moving away (receding) from us, but also that the speed at which they are moving away (the recessional velocity) is proportional to their distance distant galaxies recede faster than nearby ones. This became known as Hubbleʼs Law. Hubbleʼs initial estimates for the recessional velocity of galaxies was very high, because at the time no-one knew that there were actually several different types of Cepheid variables, with slightly different Period-Luminosity relationships. However, recent advances in astronomy have now narrowed down the value of the slope of the graph (called Hubbleʼs - 2 of 7
constant ), and results are converging to an accepted value of ~65 km/s/mpc (i.e. galaxies recede by an extra 65 km/s for every Megaparsec they are away from us). Figure 1. Hubbleʼs original plot of the recessional velocity of galaxies vs. their distance from us. (10 6 parsecs = 1Megaparsec or 1Mpc = 3.08x10 19 km) 1. a). When Hubble plotted his original graph, as shown in Figure 1 above, he made a mistake in the axes. Can you see what his mistake was? b). The equation of a straight line is given by : y=mx+c Let v be the velocities of the galaxies, and d be their distance. Write the equation for Hubbleʼs graph above. - 3 of 7
c). Using the solid black line as the line of best fit for the data in Hubbleʼs graph, calculate a value for the gradient of the line (in units of km/s/megaparsec). Galaxies - Hubbleʼs Law 2. The gradient of this line is known as Hubbleʼs constant, H0, and astronomers use this constant to find the age of the Universe: Hubbleʼs Law is given by: v = H0d (1) If a galaxy has travelled at a speed v for a time t since the Universe began, it will have travelled a distance d given by: d=vt (2) Substituting (2) into (1) gives: v=h0vt (3) and cancelling out the vʼs gives: 1 = t H 0 (4) a). Given equation 1 above what are the units for H0? b). By converting Mpc to km, equation 4 can be used to calculate the age of the Universe. If 1Mpc = 3.08 x 10 19 km, what is 1/H0 in units of seconds? - 4 of 7
c). From the value of Hubbleʼs constant that you estimated from Hubbleʼs original data (as found in Question 1 (c) ), calculate the age of the Universe in years, (Hint: 1 year = 31,556,926 seconds; 1Mpc = 3.08 x 10 19 km). d). The age of the Sun is approximately 4.6 x 10 9 years old. How does this compare with the age of the Universe as calculated from Hubbleʼs original data (Question 2 (c) above)? Do these two values make sense? Explain your answer. Going further out.. For galaxies at larger distances than those in the nearby Universe (around 10-20 Mpc), using Cepheid variables to find their distance becomes impossible, since individual stars canʼt be seen. At these distances, astronomers use other types of standard candles (objects whose brightnesses are believed to be well known). One of these is Type Ia Supernovae. Measuring distances with Type Ia supernovae When a supernova explodes, its light intensity brightens to a peak, and then falls off gradually over time. Astronomers believe that for the class of supernovae known as Type Ia (where a white dwarf star which has accreted too much mass from a companion star, explodes), this peak always reaches the same brightness. So, by measuring the brightnesses of these supernovae in distant galaxies, the inverse square law can be used to find the distance to the supernova and therefore the distance to the galaxy (see end of document for explanation on inverse-square law): b = l 4πd 2 (3) - 5 of 7
where: b = apparent brightness of the object l = luminosity of the object d = distance to the object 3. a). The table below contains modern values for the velocity and distances of galaxies using Type Ia supernovae as standard candles. Using the appropriate ICT, or graph paper, plot the following data for the velocity (y axis) and distance (x axis) of a selection of galaxies. Velocity (km/s) Distance (Mpc) Velocity (km/s) Distance (Mpc) 5420 95 19200 335 6290 108 7430 120 30300 470 15400 280 22400 385 15000 250 7870 133 8400 142 15070 255 37300 660 7230 115 7780 140 10700 181 4220 70 3520 63 12900 210 6600 115 23900 400 10570 175 26200 424 13500 229 b). Draw a line of best fit through the data points on your graph, making sure it passes through the origin (the 0,0 point) and find the gradient of this line (Hubbleʼs constant). - 6 of 7
c). From your value of Hubbleʼs constant, calculate the age of the Universe in years. d). How does this value of Hubbleʼs constant compare to Hubbleʼs original value from 1929? Inverse-square Law Luminosity is the amount of radiation emitted by an object per unit time, and this does not change with distance. However, apparent brightness, b, which we measure from Earth does change with distance - it is proportional to 1 divided by the square of the distance, or b 1 d 2 So, if you moved the object to twice as far away as its original position, its measured brightness would decrease by a factor of 4. If you moved it to three times the distance, it would decrease by a factor of 9. This is shown in the diagram below. Here, the lines represent the intensity or brightness of the source at point S. As you move further out, the intensity spreads out over the surface of the sphere. Since the sphere is bigger at larger distances (as it has a larger radius), the intensity or brightness from the source, is spread over a larger area, and so appears dimmer. The surface area of a sphere, A, is related to its radius, r, by: A=4πr 2 A star radiates over a spherical area so we can write the equation between its luminosity, L, and measured brightness, b, at distance d (which is the same as the radius of the sphere) as: b = L 4πd 2-7 of 7