Astronomy Lab - Measuring Angles and Distances in the Universe

Astronomy Lab — Measuring Angles and Distances in the Universe Introduction Since we lack the ability to travel between the stars we cannot simply measure distances between two stars or galaxies by physically traversing and logging the distance travelled, therefore we use observational measurements and techniques to determine astronomical distances. The goal of this lab is to calculate the sizes and distances to several galaxies. You will use that information later in the semester to find the Hubble constant, from which the age of the Universe can be determined. Early astronomers thought that we live at the centre of a spherical Universe, in which the sun, moon, planets and stars being attached to a giant Celestial Sphere - imagine the Earth enclosed in a hollow sphere on which the stars and planets are painted. In fact, when you look up at the sky that is just what you see - the Universe projected onto the Celestial Sphere. Astronomers still use the concept of the Celestial Sphere to find their way around the sky. Figure 1 There are just two fundamental ways to measure distances and sizes: we can compare the apparent brightness of an object with its actual energy output or we can measure the apparent (angular) size of an object as it is projected onto the Celestial Sphere and use that to find its distance or its linear size. All of our measurements of the positions and motions of objects in the sky are simply angles. As Figure 1 shows, however far away two stars may actually be from us, the apparent angle between them is independent of their distance. Stars A and B in the diagram appear to be just as far apart on the sky as stars C and D. In the same way, the apparent angular size of an object depends both on its linear size and on its distance. If we know (or assume) that two galaxies have the same physical dimensions, then we can use that information to calculate their relative distances. This situation is shown in Figure 2. If we want to find the actual distances of the two galaxies, we would need to know the distance of at least one of them by some method – then the distance to the other galaxy can be found. 1 / 11 Figure 2 Your hands and fingers are a somewhat accurate (and convenient) measuring tool. When you hold your hand at arm’s length, you can estimate angular sizes of objects, as shown in Figure 3. Figure 3: Rule of thumb from left to right: (1) The width of your little finger at arm’s length is about 1°. (2) Hold your three middle fingers together; they span about 5°. (3) Clench your fist at arms length and hold it with the back of your hand facing you; width is about 10°. (4) Stretch your three middle fingers as far as you can; the span is about 15°. (5) Stretch your thumb and little finger as far from each other as you can; the span from tip to tip is about 25°. We also use angles for measurement of time and position on the earth. For example, the angle between the direction to the sun and the celestial meridian (the North-South line that passes directly overhead) is a measure of the time since noon. A sundial is a device for measuring apparent time, as marked by the position of the Sun. For navigation, the altitude of the celestial pole (i.e., the position of the star Polaris 2 / 11 on the sky) or the direction to the Sun can help to measure the location of a ship at sea. So the concepts of this lab are simple, and perhaps not terribly exciting. But nearly everything we do in astronomy, navigation or geography depends on angular measurement. So we will start the labs with a little warm-up exercise – astronomical callisthenics, so to speak. This lab will consist of some exercises for finding angles and distances in the Universe, and for finding position on the Earth. You will start by using a sextant learn how to measure angles and to study what is called the “small angle approximation” which astronomers use when the angles involved are “small”. Whenever angles are smaller than a few degrees, simple trigonometry is possible without messy things like sines and cosines. After learning how to use a sextant, you will use it to measure your latitude. Available Resources 1. Metre sticks 2. Metric tape measure 3. Sextants 4. Celestial pipes 5. Galaxy images Methods and Procedures Part I — Distance vs. Angular Size If θ is the angular size of an object, l is the linear size, and d is the distance, simple trigonometry shows l tan θ = (1) d where angle θ is in unit of radians, and l and d have the same unit, eg. metres. When the angle is very small, we have θ ≈ sin θ ≈ tan θ (2) which is known as the Small Angle Approximation. Therefore for astronomical objects our formula is θ= l (3) d 1. Position a metre stick against the wall at one end of the lab, stand back from the metre stick, and hold your fist out in front of you. 2. Back up, looking at the metre stick down the length of your arm, until your fist just covers the metre stick and it is hidden from view. 3. Use a sextant to measure the angular size of the metre stick from your location, convert it to radians, and determine the distance indirectly using Equation 3. 4. Measure the distance directly using a metric tape measure, and calculated the percent difference based on the actual distance. 3 / 11 Trial Yours θ/° θ/rad dcalculated /m dmeasured /m % Difference Your partner’s Table 1 5. Use the hallway outside the lab to place your metre stick as far away as you can. 6. Use the sextant again to measure the angular size of the metre stick. 7. Calculate the distance twice from Equations 1 and 3 respectively. 8. Calculate the percent error based on d1 , the result without approximation, and thereby verify the validity of small angle approximation. Trial Yours θ/° θ/rad d1 /m d2 /m % Difference Your partner’s Table 2 Part II — Finding Latitude The BU observatory has a device called the “celestial pipes”, consisting of several pieces of pipe shaped to represent the celestial meridian and the celestial equator. With this arrangement, it is possible to estimate the approximate positions of stars and planets. We will use it to measure the latitude of Boston. First, recall that a great circle is the curve resulting from the intersection of a sphere with a plane passing through the centre of that sphere. To specify location of objects in the two-dimensional sky, the most straightforward coordinate system is based on the observer’s local horizon, the altitude-azimuth (or horizon) coordinate system. The point overhead directly above the observer is the zenith. The meridian is a great circle passing through the observer’s zenith and intersecting the horizon due north and south. The altitude of an object is the angle measured from the horizon to the object along a great circle, which passes through that object and zenith. The azimuth is the angle measured along the horizon eastward from north to the great circle used for the measure of altitude. Although simple to define, the altitude-azimuth coordinate system is difficult to use in practice due to its dependence on the observer’s location and Earth’s rotation causing apparent motion of stars across the sky. If stars were simply attached to a celestial sphere that rotated about an axis passing through the North and South poles of Earth, then we may define a co-moving coordinate system such that stars have constant coordinates. This rotation axis intersects the celestial sphere at the north and south celestial poles (NCP and SCP). The superior co-moving coordinate system is the equatorial coordinate system, which is based 4 / 11 Figure 4: The Celestial Sphere on the latitude-longitude system of Earth but does not participate in Earth’s rotation. Declination is equivalent of latitude and is measured in degrees north or south of the celestial equator. Right ascension is analogous to longitude and is measured eastward along the celestial equator from the vernal equinox to the object’s hour circle (the great circle passing through the object and the NCP/SCP). 9. When you stand in the middle under the celestial pipes, you can see these important circles projected onto the actual celestial sphere. The North-to-South pipe represents the celestial meridian. The point overhead directly above the observer is the zenith. 10. Identify the small crossbar on the celestial meridian. This crossbar locates the NCP. Polaris should be very close to it at night. 11. Looking in the opposite direction you should be able to identify the pipe representing the celestial equator. Note that every 10° of declination is marked on the meridian pipe, and every hour of right ascension on the celestial equator pipe. 12. To find your latitude, stand in the middle of the celestial pipes and use your sextant to find the 5 / 11 altitude of the NCP above the northern horizon. Record your result: Part III — Finding Galaxy Distances In 1912, V. M. Slipher, an astronomer at the Lowell Observatory in Flagstaff Arizona, found that most of the galaxies he was studying are moving away from us. A few years later, Edwin Hubble (after whom the Hubble Space Telescope is named), found that not only are most galaxies moving away from us, but that a very simple relationship exists between the distance to a galaxy and its recessional velocity (as measured by its Doppler shift, or redshift). This is known as Hubble’s Law v = H(z)d (4) where v is the recessional velocity of a galaxy, d is the distance to the galaxy, and H(z) is the Hubble parameter that changes over the history of the universe. H is a function of time, or redshift z. The value of H at present is (mistakenly) called the Hubble constant. The most current measurement of H0 is from ESA’s Planck mission, H0 = 67.80 ± 0.77 km/s/Mpc (5) In this exercise, you will use the concepts of this lab to measure the distances to several galaxies. In the attached collection are images of several galaxies taken by BU faculty and students using the Perkins 72-inch Telescope and PRISM camera (for “Perkins Re-Imaging System”) with Blue, Visual and Red filters at Lowell Observatory, where Slipher did his work on galaxy redshift. All of the images are reproduced at the same scale. These galaxies have been classified in the “Carnegie Atlas of Galaxies” by Allen Sandage as type Sc(s)II, which means that they are of Hubble type Sc, and Luminosity Class II. Galaxies of the same type in the Hubble Sequence have similar sizes. One possible method for getting galaxy distances is to measure their apparent angular diameters (that is, their sizes as they appear on photographs). From Equation 3 we can write θ= l d = l′ d′ (6) where l′ is the longest dimension of the galaxy measured on a photograph, d′ is some constant, and unprimed variables represent actual quantities measured in physical space. Rearranging, we derive the relationship between distance d and apparent size l′ d= ld′ (7) l′ To simplify things, we will assume the attached galaxies all have the same physical linear diameter l, since they belong to the same Hubble class. Now that both l and d′ are constants, define α ≡ ld′ to absorb the constants. Then α (8) d= l′ This inversely linear relationship is exactly what we would expect since farther objects appear smaller on photographs, and higher distance corresponds to smaller angular size. In order to calibrate this relationship, we need a reference galaxy whose d and l′ are already known. The distance of NGC 2090 was measured to be approximately 12.3 Mpc by BU students using the Hubble Space Telescope. Its angular size can be measured on the photograph. Let d0 = 12.3 Mpc and l′0 be the 6 / 11 distance and angular size of NGC 2090. Let d and l be the parameters of another galaxy. Then α = d0 l0′ = dl′ d= (9) l0′ d0 (10) l′ Note that we never need the value of α explicitly. The proportionality constant must cancel out given a reference galaxy, in our case, NGC 2090. 13. Using a millimetre ruler, measure the diameter (longest dimension) of each galaxy image in the attached collection, taking into account the faintest visible parts of the image. 14. Determine the actual distance of each galaxy, using d0 = 12.3 Mpc and your measurement of l ′0. Galaxy v/km s−1 NGC 1637 714 NGC 2090 923 NGC 4414 717 NGC 6643 1485 NGC 6801 4319 l′ /mm d/Mpc 1 /mm−1 l′ 12.3 Table 3 Analysis and Discussion 1. According to your data in Table 2, is the small angle approximation valid? 2. Why is latitude equal to the altitude of the NCP above local horizon? Explain by sketching the geometry. You are welcome to write a simple proof. 7 / 11 3. To extract the Hubble parameter from our data, plot a v–d graph (v versus d) and fit the graph with Hubble’s Law v = H0 d using Data Table 3. That is, draw the line of best fit through all data points, determine H0 , and compare your result with Planck’s value by calculating the percent difference. 4. A fancy thing to plot is v– 1 l′ graph, using the last column in Data Table 3. Is the function linear? Why or why not? 8 / 11 Appendix — Galaxy Images Figure 5: NGC 1637. v = 714 km s−1 . Figure 6: NGC 2090. v = 923 km s−1 . 9 / 11 Figure 7: NGC 4414. v = 717 km s−1 . Figure 8: NGC 6643. v = 1485 km s−1 . 10 / 11 Figure 9: NGC 6801. v = 4319 km s−1 . 11 / 11