Astronomy 153 & 154 Lab 2 Excel, Units and Conversions + Angles and Coordinates In Astronomy lab, there are important skills and concepts that students will need to use and understand in order to complete the exercises. Students need to be familiar with scientific units, and the units specifically used in Astronomy, as well as how to convert between units for calculations. Students should be familiar with angles, and their application to astronomy. Astronomers also use specialized systems of coordinates, which students need to be able to use. Spreadsheet programs like Microsoft Excel and OpenOffice Spreadsheet can simplify and repeat calculations. Many lab exercises will involve Excel to simplify calculations. Units and Conversions Prominent Radio Telescopes Around the World Image from: http://www.universetoday.com/wp-content/uploads/2009/01/worldmap-iya-780.jpg Logo of the International Astronomical Union Image from: www.iau.org Science is an international effort, and all scientists must be able to share their work between collaborations. So scientists everywhere strive to use the same units and systems of coordinates. In general, scientists use the Metric System to describe measurable quantities at a normal scale. The metric system starts with base units of gram (g) for mass, meter (m) for length, and second (s) for time. By incrementing in powers of ten to and comnbining the base units, you can form all other metric units. There are names for many powers of ten, and the most used powers are listed below. These names are prefixes to attach to the beginning of your base unit to indicate which power of ten is used.
Decimal Scientific Unit Prefix (abbreviation) 0.000000001 10-9 nano- (n) 0.000001 10-6 micro- (μ) 0.001 10-3 milli- (m) 0.01 10-2 centi- (c) 1 10 0 1000 10 3 kilo- (k) 1000000 10 6 Mega- (M) This chart shows that 1 centimeter (cm) is equal to 0.01 meters. 1 nanometer (nm) is equal to 10-9 m, and 1 Megasecond (Ms) is equal to 10 6 seconds. A historic convention that we use in Astronomy and Physics, is to use the kilogram (=1000g) as the starting unit for mass. Also note the diffenerence between capital 'M' as a prefix and lower case 'm': It's a difference of 1.000,000,000, so be careful which you use! It is frequently necessary to show measurements in units other than the ones given. Often, the formulas you will use in Astronomy are calculated only using certain units. An example of this might be if you are given a time in years, and need to use that time interval in seconds to calulate a planetary period. To convert between the units you are given to units that you want, you need a conversion factor. A conversion factor is simply a statement that two amounts in different units are equal. Statements like: there are 12 eggs in a dozen, there are 100 centimenters in a meter, and there are 7 days in a week. You put that statement in a table with the given amount in your original units and do some arithmetic! Given Quantitiy in Original Units Desired Units Conversion Factor Original Units Conversion Factor Given Quantitiy in Original Units Desired Units Conversion Factor Original Units Conversion Factor These tables show how to order the conversion factor as a fraction on the right. Multiply the Given Quantity in Original Units by the Conversion Factor Fraction. See from the conversion tables that the original units cancel out, and all that is left are the Desired units! 17 Eggs 1 Dozen 12 Eggs = 1.4 Dozen 500 gram 1 kilogram = 0.5 kilogram 1000 gram
37.4 minutes 60 seconds 1 minute = 2244 seconds 2.94 mi 1.609 km 1 mi 1000 m = 4730 m 1 km The Earth-Sun Distance Image from: http://aaaaablog.blogspot.com/2012/01/earth-at-perihelion-today.html Astronomers deal with objects and distances at such vastly different scales than normally experienced, that normal metric units can be difficult to use, even in scientific notation. For example, the Sun has a mass of 1.99x10 30 kg. That's 1,990,000,000,000,000,000,000,000,000,000 kg. Imagine having to type that into a calulator every time you needed to solve a physical problem involving the mass of the Sun. It would get very cumbersome, very quickly. Instead, astronomers use natural astronomical units to describe theses quantities. The mass of the Sun is defined to be 1 Solar Mass (Msol), and is equal to 1.99x10 30 kg. The distance that light travels in one year is called 1 LightYear (ly) and is equal to 9.46x10 15 m. The average distance between the Earth and the Sun is 1 Astronomical Unit (AU) and is equal to 1.5x10 11 m. In fact any natural measurable quantity can be given its own unit. Further examples are the Mearth for Earth Mass, Rvenus for Venus Radius and so on. There is an atached sheet of these Natural Astronomical Units and conversions to metric units. You'll need to refer to this sheet to complete this and future lab exercises. A natural question to ask at this point would be, When should I use which units? The first answer is the simplest, if the problem calls for certain units, convert to those units and use them throughout the problem. A slighlty more complicated answer is that many astronomical formulas have been put together so that they only work out correctly when quantities are given in specific units. For example. Kepler's P 2 =A 3 law only works out if the period is in Earth Years, and distance in Astronomical Units. Otherwise, there are proportionality constants that have to be tabulated with the formula.
The Andromeda Galxy as seen from the Hubble Space Telescope Image from: http://inthe-universe.blogspot.com/ The final and most complicated answer to the above question is, that the units used to describe a quantity depend on what you're trying to describe. If you want to discuss the masses of other planets, you might refer to units of Earth Masses (Mearth). If you're describing the masses of other stars, you use units of Solar Masses (Msol). For Distances inside Solar Systems, Astronomical Units (AU) are often the most reasonable. For distances between stars in a galaxy, Light Years (ly) are the most used units. Finally, for distances between galaxies, and to talk about larger scale structure in the Universe, the most frequently used unit is the parsec (pc). Angles & Coordinates Angles can be measured in degrees, minutes, and seconds or in radians. There are 360 o in a full circle or 2π radians. There are 60 minutes in 1 degree of angle, and 60 seconds in 1 degree. To distinguish between minutes or seconds in angle and time, Astronomers refer to arcminutes and arcseconds as fractions of angles in the sky. The notation is o degrees, 'minutes, ''seconds. Radians are the general scientific units of angle, but it is much more common in astronomy to use degrees, minutes, and seconds. Excel, however, uses radians. So all calculations with angles in Excel must first convert angles in o, ', '' to decimals of degrees, and then convert degrees to radians for the program. Converting angles in o, ', '' to decimal uses exactly the same method as any other unit, and the same factors as time. 14 min of arc 1 degree 60 min of arc = 0.23 degrees 45 degrees 2π radians 360 degrees = 0.785 radians = π/4 radians
The Celestial Sphere Image from: http://www.castlerock.wednet.edu/hs/stello/astronomy/text/chaisson/bg30p/html/bg30p02.htm The earth is divided up into lines of latitude and longitude to find any location on the planet. If you extend these lines to project on the sky, they become Declination (Dec) and Right Ascension (RA). The Right Ascension and Declination system of celestial coordinates is used to chart and find objects on the celestial sphere- our projection of the night sky. Right Ascension and Declination are fixed on the celestial sphere, so that even as the Earth rotates under the stars, the celestial objects keep the same RA and Dec coordinates. Declination is measured in degrees North or South of the Celestial Equator, a projection of Earth's Equator. So any point on the Celestial Equator is at 0 o Declination. The North Celestial Pole is at +90 o Dec (North), and the South Celestial Pole is at -90 o Dec (South). Right Ascension is measured in angular units of hours, minutes, and seconds because RA is in the orientation of the Earth's rotation. There are 24 hours in one day, which is one Earth revolution, which is 360 o of rotation. So 1 hr = 360 o /24hr=15 o. In Earth's longitude, the Prime Meridian is arbitrarily taken through the city of Greenwich England. Similarly, the zero line for Right Ascension, called the Celestial Meridian, is arbitrarily chosen to be the RA of the Sun at the instant of the Vernal Equinox. Right Ascension then increases from 0h to 24h Eastward from the line through that point. With this set of Right Ascension and Declination Coordinates, every object in the night sky can be given a fixed set of coordinates so that Astronomers can find the object again every night. On the other hand, it is sometimes more convenient to indicate the position of a celestial object as it appears when it is seen. For a more impromptu coordinate sytem, astronomers turn to Altitude and Azimuth. Altitude is the height of an object above the horizon, from 0 o at the horizon, to 90 o at the zenith, or point directly overhead. Azimuth is the angle in degrees from 0 o to 360 o starting from North going in a circle through East, South, West, then back to North. The cardinal directions would have Azimuths in steps of 90 o. North is at 0 o Azimuth, East at 90 o, South at 180 o, West at 270 o, and back up to 360 o at North again. In this way, every object visible in the sky will have an Altitude and Azimuth Coordinate. But the A&A coordinates change rapidly over the course of a night, as the Earth rotates, throughout the year as the Earth orbits the Sun, and varies depending on location. So A&A can be used to point out
objects in the sky in-the-moment. This is the Altitude that the Lunar Observation Lab refers to when it asks you to write the Altitude of the moon at the time you observe and sketch it. Diagram of Altitude and Azimuth Image from: http://crab0.astr.nthu.edu.tw/~hchang/ga1/ch02-02.htm Excel Excel is a useful tool for completing repetitive, complicated, and interrelated calculations. You can make Excel perform the same arithmetic on a large set of numbers, plot two sets of numbers to see a correlation, and manipulate large selections of data. In the Lab Exercises, we will be using Excel frequently to reduce the amount of pencil work in the class. You should complete the Lab2WS by hand before performing the Excel calculations. Your TA will demonstrate the first calculations for you on the screen and you will complete the exercise on your own in Excel by typing in the formulas and selecting the appropriate cells to perform the calculations. For this exercise, an Excel template is provided, in addition to the list of convenient conversion factors. Open up this Excel document by going to the Astrolab website, clicking on the template link, click 'save as' and save the document to the desktop. Now you should be able to open the Excel template from the desktop by double clicking on the icon. When the document opens, you will notice that many of the columns have already been filled in. There are already values for planetary mass in kg, orbital radii in km, and orbital period in Earth days, as well as the mass in solar masses and distances in light years for several prominent stars, and finally the distance in kiloparsecs for several notable galaxies. You'll note that next to these filled in columns are blank columns with no numbers. It is your goal in this exercise to fill in those columns by converting the values in the given units, to the units at the top of each blank column. To make Excel perform a function on a value contained in a cell, you must first select the empty cell to hold the new value with a left mouse click. Then press the = key. The beginning of every formula in Excel is an = sign. The = sign tells Excel that the formula should be calculated. Now, you need to enter the calculation into the cell. Click on the cell whose value you want to convert, a number and letter should appear in the empty box corresponding to the column and row of the selected value. Now enter your arithmetic symbol, in this case * for multiplication, since you want to multiply the value form the filled column, by the conversion factor. Put the conversion factor in parentheses, without typing out the units! The parentheses tell Excel to do the arithmetic of that
portion seperately from the whole calculation. If you were to type in the units, like you wrote them down to show cancellation, Excel would not understand, so just type in the numbers for the unit conversions. On the worksheet provided, you should write out the steps for all the conversions in general showing how the units cancel. After you have entered the conversion factor, close the parentheses and press the Enter key. When you press the Enter key, Excel then should tabulate your formula and give a value in the formerly blank box. One of the most useful features of Excel is that you don't need to type in an identical formula many times. For conversions in a column that require the same conversion factor, move your cursor over the bottom right corner of the newly completed cell. A cross-hairs should appear. Click and hold that corner and drag it down the column until you reach the end of that unit conversion. When you release the cursor, Excel will have tabulated all the values in that column! Complete the Excel table by entering in the appropriate unit conversions into each eampty column, and dragging the corner down to repeat identical calculations. Print out the full Excel sheet on the Lab printer and attach it to your completed Lab Worksheet.
Name Section Number Date TA Name Astronomy 153 & 154 Lab 2 Worksheet Excel, Units & Conversions, Angles & Coordinates Unit Conversions: Show all work! 1. What is 0.35 km in units of meters? 2. What is 133 days in units of seconds? 3. How many minutes are in 27 years? 4. What is the mass of the Sun in Megagrams? (Msol=1.99x10 30 kg) 5. The distance between the Milky Way and Andromeda Galaxy is 2.54 Mly. What is that distance in Mpc? 6. The star Betelgeuse is about 197 pc away. How far away is Betelgeuse in ly? In meters? Angles and Coordinates 7. How many radians are in 270 o? 8. What is the decimal angle in degrees and radians, for the angle 47 o 23'33''?
9. What is the celestial analogue of latitude? Longitude? Specify which! 10. The Pole Star, Polaris, has a RA: 2h31m49s, Dec:+89 o 15'51''. What is the angle in degrees represented by Polaris' RA? 11. What is the Altitude and Azimuth of the pencil sharpener in the astronomy classroom, if Ayres Clock Tower points due North? 12. What is the Altitude and Azimuth of the top lintel of the Astronomy classroom's doorway? Excel For problems 13-19, explicitly write out the unit conversions you entered into Excel for the empty columns. Indicate how the units cancel to give the final result, but do not carry out each individual calculation by hand, that's why we used Excel!