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Exp 2B Hubble UNIVERSITY OF SURREY DEPARTMENT OF PHYSICS Level 1: Experiment 2B THE HUBBLE CONSTANT AND PULSAR RADIO ASTRONOMY 1 AIMS 1.1 Physics This experiment consists of a pair of computer simulations that mimic ‘virtual’ telescopes. The first part of the experiment simulates a robotic optical telescope equipped with a spectrometer to examine the spectra of galaxies and to measure their apparent magnitudes, whilst the second part simulates a radio telescope to measure the MHz frequency signals from neutron star pulsars. The specific aims of the two parts of the experiment are as follows: Part 1: Hubble Constant and Red-Shift Measurements. Spectrometer data is used to measure the spectra of light emitted from galaxies, and their apparent magnitudes. Using Pogson’s law you will determine their distances from the Earth. From the spectra you will use the Doppler Effect to determine their velocities. At the end of the first part of this experiment you should be able to: l Understand how the position of an object in space can be determined by a co- ordinate system based on a sphere, and understand the parsec as a unit of astronomical distance. l Understand how stellar magnitudes can be used to estimate astronomical distances. l Calculate a value for the Hubble constant and the rate at which the universe is expanding. l Estimate the age of the universe and assess some of the problems in obtaining a reliable estimate. Part 2: Pulsar Radio Astronomy. The periodic radio signals received from neutron star pulsars are examined, and the period of the signal is measured. By studying the variations in the arrival time of pulsar signals as a function of the signal frequency, the dispersion of electromagnetic radiation travelling through space is measured. From these dispersion measurements, the distance of a typical pulsar can be calculated. At the end of this part of the experiment you should be able to: l Understand the nature and mechanism of pulsar signals from neutron stars. l Measure the period of a pulsar signal and show that the signal strength varies as a function of radio frequency. l Understand the effect of dispersion in the propagation of pulsar signals, and use this effect to measure the distance of a typical pulsar from earth. Last updated on 4/2/05 by PJS Experiment 2B - Hubble Constant and Pulsars 1.2 Skills The particular skills you will start to acquire by performing this experiment are: l The ability to use a realistic computer simulation. l Appreciate the use of optical and radio telescopes to acquire spectra and radio signals, and how this data can be used to determine stellar and extragalactic distances. l Researching the scientific methods for measuring stellar and pulsar distances and the latest findings on the expansion and age of the universe. PART1: THE HUBBLE CONSTANT - REDSHIFT RELATIONSHIP 2. INTRODUCTION AND THEORY The instructions for using the computer simulation are contained in Appendix A. There are some terms and expressions however that you may be unfamiliar with. These are explained below: 2.1 The Celestial Coordinate System Right Ascension and Declination. This is a celestial coordinate system similar to the terrestrial system of longitude and latitude with the earth situated at the centre of a celestial sphere. Declination (symbol d or Dec) is analogous to latitude and is the angular distance of a celestial object above or below the celestial equator. It is measured in degrees, seconds and minutes of arc as shown in Figure 1. Right Ascension (symbol RA or a) is analogous to terrestrial longitude and is the angular distance between the hour circle though an object and the hour circle through the vernal equinox. RA is usually measured in hours, minutes and seconds. In this way the position of a celestial object can be uniquely specified by its RA and Dec as shown in Figure 1. Figure 1: The terrestrial coordinate system of longitude and latitude (left), and the celestial coordinate system of Right Ascension and Declination (right). Page 2 of 18 Experiment 2B - Hubble Constant and Pulsars 2.2 Absolute and Apparent Magnitudes – Pogson’s Law & Stellar Distances The magnitude scale on which stars are rated is based on a convention first devised by Hipparchus. They are classified by apparent magnitude by rating the brightest stars that can be seen with the naked eye as magnitude 1.0, and the faintest as magnitude 6.0. Later, with the invention of the telescope and for consistency, it became necessary to assign to some stars a magnitude brighter than 1.0. Hence Vega is magnitude 0 and Sirius, magnitude -1.4. The Sun has an apparent magnitude of -26.74! It is important to understand that the more negative the apparent magnitude, the brighter the star appears. Or conversely, the larger the magnitude (more positive), the fainter the star appears. To understand why this is so we need to note two things: The human eye perceives equal ratios of brightness at equal intervals. For example, if you tried to compare the relative brightness of a 100W and 200W light bulb at the same distance, then the 200W bulb would not appear twice as bright as the 100W bulb. What you would observe however, is that the difference in brightness between the 100W and the 200W bulb will appear the same as the difference in brightness between a 200W and a 400W bulb. On Hipparchus’s scale, the flux (W.m-2) coming from stars of the 1st magnitude was about 100 times greater than that from stars of the 6th magnitude. A difference of 6 - 1 = 5 magnitudes therefore, corresponds to a flux ratio of 100. This means that a magnitude difference of 1 corresponds to a flux ratio of (100)1/5 or 2.512. Pogson in 1856, formulated Hipparchus somewhat subjective scale into a precise mathematical law: æ f ö ç 2 ÷ m2 - m1 = -2.5logç ÷ (1) è f1 ø where m1 and m2 are the apparent magnitudes of star 1 and star 2, respectively and f1 and f2 are the received flux from star 1 and star 2, respectively. We can see that equation (1) is sensible if we note that the minus sign ensures that magnitudes are a measure of faintness. If f2 < f1 (star 2 is fainter than star 1) then the flux ratio is < 1 and the log of the ratio is negative. So m2 - m1 is positive and the fainter star has the larger magnitude as expected. The multiplier 2.5 is a scaling factor that ensures that a flux ratio of 100 corresponds to a magnitude difference of 5. e.g. if f2 / f1 = 1/100 then m2 - m1 = 5. We can now use Pogson’s law to derive an absolute magnitude scale and use it to estimate stellar distances. Suppose you could place all the stars at a fixed distance from the earth. The differing distances would not then be a factor in how bright the stars appeared. Instead, the differences in magnitude would be only due to differences in luminosity and as such, these values would be absolute. Astronomers use a standard distance of 10 parsecs for absolute magnitude comparison. See Appendix C for explanation of the parsec (pc). The magnitude that a star would have 10pc from the earth is thus defined as its absolute magnitude. If a star has a Luminosity L with an absolute magnitude M then assuming that the luminosity is radiated uniformly over the area of a sphere, the flux received at the earth if the star is at a distance r is Page 3 of 18 Experiment 2B - Hubble Constant and Pulsars L f = (2) 4pr 2 Let f10 be the flux the star would have at a distance of 10pc. Then from Pogson’s law, 2 2 2 2 m - M = -2.5 log (f / f10) where f10 = L / (4p.10 ) so f/f10 = 10 /r = (10/r) . We can thus write Pogson’s law as: 2 æ10 ö m - M = -2.5logç ÷ = -5log10 + 5log r è r ø or m - M = 5log r - 5 (3) In the experiment you will use equation (3) by measuring the apparent magnitude m and assuming a value for the absolute magnitude M in order to calculate r. For the purposes of the experiment, the absolute magnitude of all galaxies that you will observe is assumed to be -22. Note that we are measuring visual magnitudes as opposed to bolometric magnitudes. i.e. the magnitude measured over all wavelengths of the electromagnetic spectrum. Pogson’s law is a reasonable one, since with the exception of very hot and very cool stars, most starlight is emitted in the visual part of the spectrum. 2.3 Doppler shift of the H & K lines of Ca The Doppler effect is something you should be familiar with. We use the relation: Dl Vr = (4) l0 c where Dl = lobserved - l0 where l0 is the wavelength observed in the laboratory and Vr is the radial velocity of the galaxy. The H and K lines of Ca are prominent spectral absorption lines of ionised calcium from stars with surface temperatures similar to the sun. The K line (393.4nm) and the H line (396.8nm) are easily observable in galaxies. Note that the spectrometer measures the wavelength in angstroms (1 Angstrom = 1Å = 10-10m). By measuring lobserved for each of the K and H lines you can use equation (4) to calculate the radial velocity in each case. You should take the average of the two values you have calculated for your measure of the radial velocity of the galaxy.
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