KEELE UNIVERSITY SCHOOL OF CHEMICAL AND PHYSICAL SCIENCES Year 1 ASTROPHYSICS LAB. WEEK 1. Introduction

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1 KEELE UNIVERSITY SCHOOL OF CHEMICAL AND PHYSICAL SCIENCES Year 1 ASTROPHYSICS LAB WEEK 1. Introduction D. E. McLaughlin January 2011 The purpose of this lab is to introduce you to some astronomical terms and definitions that you will encounter in the rest of this lab course; to refresh your memory about some basics of error propagation in calculations involving experimental data; and to give you practice using Excel to aid these calculations and to fit straight lines to data. Answer all of the questions below in your lab book, and get a demonstrator or a member of staff to check that your work is complete and correct preferably, as you go. You should make your work clear enough and detailed enough to serve as a useful reference for yourself during the later labs. 1. Parallax and Angular Distances Read about trigonometric parallax at Nick Strobel's website ( so that you understand the figure below, which is taken from his site. Based on what you have read, answer the questions following. Figure 1: Trigonometric parallax. Drawing copyright by N. Strobel (astronomynotes.com). 1. Given that 1 Astronomical Unit is equal to 1.5x10 11 m, that there are 3600 arcseconds in one degree, and that there are 2 radians in 360, calculate the value of one parsec (1 pc) in units of metres, AU, and light years. (Use the small-angle approximation: sin(x) tan(x) x, for x 1 in radians.) 2. A star is at a distance of 6 pc from the Sun. What is its trigonometric parallax (in arcsec)? 3. Sketch a diagram analogous to Figure 1, but in which an observer on Earth views two stars

2 at a common, known distance D from the Earth. The stars are separated from each other by an unknown distance s (not 1 AU!), and their angular separation on the plane of the sky is θ. Find a formula for s in terms of D and θ. 4. Two objects on the plane of the sky have an angular separation of 206 arcsec. What is the distance between them in km, if they are both 2 AU away from Earth? (What might such objects be?) 5. Two stars in a galaxy are separated from each other by 8 kpc. If both stars are 20 Mpc distant from the Sun. What is the angular distance between the stars (in arcsec) measured by an observer on Earth? 2. Celestial Coordinates Read about celestial coordinates at Nick Strobel's website ( and answer the following: 6. Write brief definitions of right ascension and declination, in your own words. 7. Two points on the celestial equator have RA=3 h 50 m 04 s and RA=4 h 02 m 00 s. What is the angular distance between the points in degrees? 8. Two points on the plane of the sky have the same RA, and declinations 66 15'55" and 65 07'10". What is the angular separation of the points in arcsec? If the "points" are two galaxies, both 100 Mpc away from the Earth, what is the distance between the galaxies in pc? 3. Magnitudes and Distance Modulus The apparent magnitude m of a star (or an entire galaxy, etc.) with an intrinsic luminosity L (measured in units of the solar luminosity), and at a distance D (in units of pc) from an observer, is a measure of its flux through a sphere of radius D, on a base-10 logarithmic scale: m = constant 2.5 x log(l / 4 π D 2 ) where the constant in this definition depends on the wavelength at which the star is observed (since stars emit different amounts of light at different wavelengths, and the transmission properties of the filters used to observe stars also depend on wavelength). The apparent magnitude of an object is often also denoted by a letter representing a standard photometric filter. Thus, V signifies an apparent magnitude measured through the standard (Johnson) V filter; a B- band magnitude can be denoted simply by B; and so on. You should do some background investigation into stellar spectra and photometric filters, if you are unfamiliar with these things. For example, see the notes and the excellent Blackbody Simulator at The absolute magnitude M of an object is the apparent magnitude it would have if it were at a distance of exactly 10 pc from an oberver. An absolute magnitude in the standard V filter is usually denoted M V, and similarly for other filters. The colour of a star is the difference between its measured (apparent) magnitudes at two different wavelengths or through two different filters. Thus, for example, B V and M B M V denote colours. 9. Two stars have absolute magnitude M V,1 = and M V,2 = Show that

3 M V,1 M V,2 = 2.5 log(l 1 / L 2 ). Thus, which of the two stars is brighter? What is the ratio of their luminosities? 10. The absolute magnitude of the Sun in the V-band is What is the absolute V-band magnitude of a star with 100 times the luminosity of the Sun? Of one with 0.01 times the luminosity of the Sun? 11. Show that the difference between the apparent magnitude of an object at a given wavelength, and its absolute magnitude at the same wavelength, is given by m M = 5 log D 5. where D is the distance to the object, measured in pc. The quantity (m M) is called the distance modulus. 12. What is the apparent V-band magnitude of a star with the luminosity of the Sun that is at a distance of 5 pc from us? At a distance of 50 pc? 13. A galaxy is known to have an absolute V-band magnitude of 20, but is measured to have an apparent V magnitude of +11. What is the distance to the galaxy? 14. Suppose a star has a colour M B M V =1.0. Is the star brighter in the B band or the V band? What is its apparent colour, B V, if it is at a distance of 5 pc from us? At a distance of 10 pc? At a distance of 30 pc? 4. Measurement Uncertainties Review the notes online about measurement uncertainties and error propagation (i.e., how to calculate the uncertainty in a function of a quantity with a measurement error), and answer the following: 15. A star has trigonometric parallax p=0.100" ± 0.003". Find the distance to the star (in pc) and the uncertainty in the distance. 16. A star has a luminosity 10±1 times the luminosity of the Sun. What is the uncertainty in its magnitude? 17. The relativistic Doppler shift relates the observed wavelength, λ, and the rest-frame wavelength, λ 0, of light emitted by an object moving at velocity V relative to an observer: V c = 0 0 Suppose light from a source is known to have rest wavelength λ 0 = 450 nm but is detected at λ = 495±15 nm. Find the speed (including uncertainty) of the source in this case. 18. A right triangle has sides a=3.0±0.2 m and b=4.0±0.3 m. What are the length of the hypotenuse, c=(a 2 +b 2 ) 1/2, and its uncertainty? 5. Excel Plotting and Fitting Straight Lines The purpose of this exercise is to get used to entering data into an Excel spreadsheet; to see how formulae can be used in Excel, both in a general sense and in the specific context of fitting straight lines to data; and to gain some practice in formatting graphs in Excel.

4 Download the spreadsheet ''Line_Fitting_Template.xls'' from the Astro Lab web page (it is the second-last of the links collected under "Key Resources"). You may well want to use this throughout your lab work this semester, to calculate and plot the best-fitting straight lines to data, so save it somewhere convenient. Then make a copy of the template so you have a file you can work on for this exercise. Open the spreadsheet. You can enter data into Columns B, C, and D of Rows 1 38; the other columns in these rows are reserved for the results of calculations using Excel formulae, which you will look at more closely below. All of Rows consist of cells with more Excel formulae in them. Below these rows is a chart, which is set up to plot the data you enter in the spreadsheet, and the best-fitting straight line to the data. The following data represent the velocities at which galaxies at various distances are moving away from us as a result of the expansion of the Universe. The aim here is to make a plot of velocity versus distance and find the slope and intercept (with uncertainties) of the straight line that best fits the data. Thus, enter these numbers into Columns B, C, and D of the first several rows of your Excel spreadsheet (you want to plot Velocity against Distance, so Distance is X the horizontal axis on the graph and Velocity is Y the vertical axis). Do not edit any cells in any other columns, or in any rows below Row 38. Distance Velocity Uncertainty Distance Velocity Uncertainty (Mpc) (km/s) (km/s) (Mpc) (km/s) (km/s) When you have entered all of the data, you should see that all of the cells in the other columns of the rows containing data have been filled in with numerical values. You should also see that the data (and errorbars) you've entered have appeared on the graph. Be sure you understand how the cells have filled out: click on several different cells to see the formulae associated with them. Also be sure you understand why these formulae are there: they are calculating the terms needed to form several different sums needed to calculate the slope and the intercept of the best-fit straight line. At this point, review the theory of straight-line fitting to data the method of least squares on pages of your Lab Manual from Semester 1; see especially equations (12) and (13) on page 20. Click on the cells containing numbers in Rows 39, 41, 42, and 43 of your spreadsheet and convince yourself that you understand how these relate to the three equations (12). Then do the following: 19. In Cell B45 of the spreadsheet, enter an Excel formula to calculate the uncertainty in the slope of the best-fit line. In Cell B46, enter an Excel formula to calculate the uncertainty in the intercept of the best-fit line. Note these formulae, and the values of the uncertainties, in your Lab book.

5 20. Change the x-axis label on the graph to read ''Distance (Mpc)'' and change the y-axis label to read ''Velocity (km/s)''. Make a trend line appear on the graph, and display the equation of the trend line on the graph. Compare this to the best-fit slope and intercept calculated on Rows 42 and 43 of the spreadsheet. Print the spreadsheet and fix it in your Lab notebook before you hand it in. Useful Reference Material Keele Astro Lab webpage Keele Physics Lab Notes (Year 1, Semester 1), Part II Strobel, N., Astronomy Notes Zombeck, M. V., 1990, Handbook of Space Astronomy and Astrophysics, 2nd ed., (Cambridge:CUP). pp