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Units, Conversations, and Scaling Giving Numbers Meaning and Context Author: Meagan White; Sean S

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Units, Conversations, and Scaling Giving Numbers Meaning and Context Author: Meagan White; Sean S Units, Conversations, and Scaling Giving numbers meaning and context Author: Meagan White; Sean S. Lindsay Version 1.3 created August 2019 Learning Goals In this lab, students will • Learn about the metric system • Learn about the units used in science and astronomy • Learn about the units used to measure angles • Learn the two sky coordinate systems used by astronomers • Learn how to perform unit conversions • Learn how to use a spreadsheet for repeated calculations • Measure lengths to collect data used in next week’s lab: Scienctific Measurement Materials • Calculator • Microsoft Excel • String as a crude length measurement tool Pre-lab Questions 1. What is the celestial analog of latitude and longitude? 2. What quantity do the following metric prefixes indicate: Giga-, Mega-, kilo-, centi-, milli-, µicro-, and nano? 3. How many degrees is a Right Ascension “hour”; How many degrees are in an arcmin? Arcsec? 4. Use the “train-track” method to convert 20 ft to inches. 1. Background In any science class, including astronomy, there are important skills and concepts that students will need to use and understand before engaging in experiments and other lab exercises. A broad list of fundamental skills developed in today’s exercise includes: • Scientific units used throughout the international science community, as well as units specific to astronomy. • Unit conversion methods to convert between units for calculations, communication, and conceptualization. • Angular measurements and their use in astronomy. How they are measured, and the angular measurements specific to astronomy. This is important for astronomers coordinate systems, i.e., equatorial coordinates on the celestial sphere. • Familiarity with spreadsheet programs, such as Microsoft Excel, Google Sheets, or OpenOffice. Spreadsheets can organize large sets of calculations and are able to simplify repeated calculations. Many of our lab exercises require use of spreadsheets. 2. Units and Conversions 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. A unit tells you what you are measuring (length, time, mass, etc.), and it is what gives a number meaning. In general, scientists use the metric system to describe measurable quantities. The metric system starts with base units of gram (g) for mass, meter (m) for length, and second (s) for time. By increasing or decreasing in powers of ten and combining base units, you can form all other metric units. The metric units defined by powers of ten are named using a prefix attached to the base unit. Examples of some of the most common named prefixes in the metric system are given in Table 1. Table 1. Metric System Prefixes Decimal Power of Ten 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) 1 100 1,000 103 kilo- (k) 1,000,000 106 Mega- (M) 1,000,000,000 109 Giga- (G) Table 1 shows that 1 centimeter (cm) is equal to 0.01 meters (m). 1 nanometer (nm) is equal to 10-9 m (0.000000001 m,) and 1 Megasecond (Ms) is equal to 106 m (1,000,000 s). While the gram (g) is the base unit for mass, astronomy and physics often uses the kilogram (1 kg = 1000 g) as the starting unit for mass. The difference between the kilogram and the gram is a factor of 1,000, so be careful which unit you are using. It is frequently necessary to show measurements in units other than the ones measured in an experiment or given to you in a problem. Often, the formulas you will use in this astronomy lab are calculated using specific units. This is because many of the so-called universal constants and other coefficients have specific units for the value given to you. Recall that a number, like a coefficient in an equation, is just a number; it is the assignment of units that gives that number meaning. A typical example of this is you are given a time in years, and you need to use that time interval in seconds to calculate a planetary period using Kepler’s Third Law of Planetary Motion. A more complicated example of this is the universal gravitational constant, G (usually said “big Gee”), which is 6.67 x 10-11 N m2 kg-2, so you must use force units in Newtons, distance units in meters, and mass units in kilograms. To convert between the units you are given to units that you want, you need to use a conversion factor. A conversion factor is simply a statement that two amounts in different units are equal. For example, there are 12 eggs in a dozen eggs (12 eggs = 1 dozen eggs), there are 100 cm in one meter (100 cm = 1 m), and there are seven days in one week (7 days = 1 week). Using the 100 cm = 1 m conversion factor as an example, let us examine how you can arrange the conversion factor as a fraction equal to 1. 100 cm = 1 m can be written in fraction form with !"" $% ! % or . Note that both of these fractions are equal to 1 since they are the same ! % !"" $% amount of distance. To use a conversion factor, you can use what is called the “train tracks method” demonstrated below. You start with the given amount in the original units, apply your conversion factor in fraction form such that the denominator cancels out the original units, and simplify using arithmetic. Conversion Factor Given Quantity in Original Unit Desired Unit = Converted Answer in Original Unit Desired Unit This conversion method requires that you arrange your conversion factor in fraction form such that the original units “cancel out.” Multiply the Given Quantity by the Conversion Factor in the correct fraction form, and you get a converted answer in your desired units. Four examples of this conversion are given below: 17 Eggs 1 Dozen 500 grams 1 kilogram = 1.4 Dozen = 0.5 12 Eggs 1000 eggs kilograms grams 37.4 minutes 60 seconds 2.94 miles 1.609 km = 2,224 s = 4.73 km 1 minute 1 mi Often, multiple conversions need to be made to get to the desired unit. The method is the same, but it requires extra steps. You start with your original unit, multiply it by a first conversion factor to convert to a “linking’ unit, and then multiple again by a second version factor to go from the linking unit to the desired unit. In this “chained conversion method,” the train-tracks method still works, but you add more “track.” Examples are given below Chained Conversion Method Example Orig. Linking Desired 5000 0.62 1 km 5000 ∗ 0.62 mi Unit Unit Unit m mi = 1000 1000 Orig. Unit Linking Unit 1 km = 3.1 mi m Chained Conversion Method -Multiple Links Original Unit Linking Unit 1 Linking Unit Linking Unit 3 Desired Unit Converted 2 = Answer in Orig. Unit Linking Unit Linking Unit 2 Linking Unit Desired Unit 1 3 Chained Conversion Method – 3 Links - Example 1 yr 365.25 days 24 hours 60 min 60 sec = 3.16 x 107 seconds 1 yr 1 day 1 hour 1 min Astronomers deal with objects and distances at such vastly different scales than our normal, everyday human experience, that regular metric units can be difficult to use, even in scientific notation. For example, the Sun has a mass of 1.99 x 1030 kg, or written out, 1,990,000,000,000,000,000,000,000,000,000 kg. Imagine having to type that into a calculator every time you needed to solve a problem involving the mass of the Sun! It would get very cumbersome, very quickly, and it would be prone to user input error. Instead, astronomers use natural astronomical units to describe certain quantities. • The mass of the Sun is defined to be 1 Solar Mass (MSun), and it is equal to 1.99 x 1030 kg. • The distance that light travels in one year is called 1 light-year (ly), and it is equal to 9.46 x 1015 m. • The average distance between the Earth and the Sun is defined to be 1 Astronomical Unit (AU), and it is equal to 1.5 x 1011 m. In fact, any natural, measurable quantity can be given its own unit. Further examples are MEarth for the mass of Earth, RJupiter for the radius of Jupiter, and so on. One RMoon would be the radius of the moon, which is 1.737 x 106 m. Table 2 summarizes some of the most commonly used natural astronomical units. Table 2. Astronomical Units Astronomical Unit Abbreviation Equivalent Unit(s) 1 Astronomical Unit AU 1.50 x 1011 m 1 light-year ly 9.46 x 1015 m 1 parsec pc 3.09 x 1016 m = 3.26 ly 1 kiloparsec kpc 3.09 x 1019 m = 1,000 pc 30 1 Solar Mass MSun 1.99 x 10 kg 8 1 Solar Radius RSun 6.96 x 10 m 26 1 Solar Luminosity LSun 3.83 x 10 W 24 1 Earth Mass MEarth 5.97 x 10 kg 6 1 Earth Radius REarth 6.37 x 10 m A question on your mind at this point might 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 slightly more complicated answer is that many astronomical formulae have been put together so that they only work out correctly when the quantities are given in specific units.
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