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1 APPENDIX A: Fundamental Quantities

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1 APPENDIX A: Fundamental Quantities 1 APPENDIX A: Fundamental Quantities There are various ways to describe the world in which we live. Some are qualitative and others are quantitative. Qualitative descriptions describe aspects of objects or events such as texture, and use words like `rough', `smooth', ‘flat’ etc. Qualitative descriptions cannot be described numerically. One would not say that you looked tired with a value of 3.0 unless someone had first set up some kind of numerical scale to measure just how tired you were; tiredness is not something we measure quantitatively. On the other hand, length is a dimension that can be described either qualitatively or quantitatively; one can qualitatively describe an object as long, or one can quantitatively describe it as 10 feet in length. All quantitative measurements are made in some kind of unit. Length, for example, can be measured in units of meters, feet, miles, etc. Other fundamental metric units are the kilogram (a measure of mass) and the second (a unit of time). Other units of measurement are combinations of these fundamental units. An example of a combination is velocity, expressed in units of meters per second (m/s) which measures how far something has moved in a given direction over a given period of time. In astronomical studies, one sometimes uses units which express rather large values in the fundamental metric units. An example of this would be the Solar Mass unit (notated as M ). The mass of our Sun is, by definition, one Solar Mass or about 1,900,000,000,000,000,000,000,000,000,000 kilograms. A star with 10 times as much mass can be written as 10 M ; this is clearly more convenient to write than a number with all those zeros! Other units used in astronomy are the light year (ly), parsec (pc), and the astronomical unit (A.U.), all of which are units of distance. The unit you choose to use depends on the situation, and personal preferences. When describing distances in the solar system the astronomical unit is typically used since it is the average distance from the Sun to the Earth. In describing distances to stars the parsec or light year is usually used. As described in the introductory lab, the metric system allows easy expression of large multiples of the fundamental units via prefixes. For example, 1,000 meters is called a kilo- meter and is usually written as 1km. As described in section 1.4 scientists also use a notation system called scientific notation for representing very large or very small numbers without having to write lots of digits. As an example of how large numbers can get in science let's look at the mass of Mars. Using Kepler's laws of motion to study Mars' moons, astronomers have determined that Mars has a mass approximately equal to 640,000,000,000,000,000,000,000 kilograms. Now you can see that it is rather inconvenient to write down all those zeroes, and it is confusing to use the prefixes above. Imagine how much more mass there is in the Galaxy and you can see that we need an easy way to write very big (or very small) numbers. This leads us to the concept of Scientific Notation. 1 2 APPENDIX B: Accuracy and Significant Digits The number of significant digits in a number is the number of non-zero digits in the number. For example, the number 12.735 has five significant digits; the number 100 has 1. When computing numbers, people today often use calculators since they give us precise answers quickly. Unfortunately, many times they give us answers that are unnecessarily and some- times unrealistically precise. In other words, they give us as many significant digits as can fit on the calculator screen. In most cases, you will not know the numbers you are plugging in to the calculator to this precise of a value, and therefore will get an answer that has too many significant digits to be correct. This will be the case for your astronomy labs this semester. In general, you should only report the accuracy of a calculation with the number of significant digits of the least certain (smallest number of significant digits) of any of the numbers which were the input into the calculation. For example, if you are dividing 13.2 by 6.8, although your calculator gives 1.94117647, you should only report two significant figures (i.e. 1.9), since that is the number of significant figures in the input number 6.8. 2 3 APPENDIX C: Unit Conversions Very often, scientists convert numbers from one set of units to another. In fact, not only do scientists do this, but you do it as well! For example, if someone asks you how tall you are, you could tell them your height in feet or even in inches. If someone said they are 72 inches tall, and 12 inches equals 1 foot, then you know they are 6 feet tall! This is nothing more than a simple conversion from units of inches to units of feet. Another everyday unit conversion is from minutes to hours, and vice versa. If it takes you 30 minutes to drive from Las Cruces to Anthony, then it takes you 0.5 hours. We know this because 60 minutes are equal to 1 hour. However, how can we write these unit conversions, with all the steps, so that we are sure we are converting units correctly (especially when the units are foreign to us (i.e. parsecs, AU, etc.))? Let us begin with our everyday conversion of inches to feet. Say a person informs you that they are 72 inches tall and you want to know how many feet tall they are. First, we need to know the unit conversion from inches to feet (12 inches = 1 foot). We then write the following equation: 1 foot 72 inches × = 6 feet (1) 12 inches Note how the inches units cancel (one in the numerator and one in the denominator) and the units which remain are feet. As for the mathematics, simply use normal rules of division (72/12 = 6) and you wind up with the correct result. The second example, minutes to hours, can be performed using the method above, but what if someone asked you how many days there are in 30 minutes? You will need to use 2 unit conversions to do this (60 minutes = 1 hours, 24 hours = 1 day). Here is how you may perform the unit conversion: 1 hour 1 day 30 minutes × × ≈ 0:0208 days = 2:08 × 10−2 days (2) 60 minutes 24 hours Again, note that the minute units have cancelled as well as the hour units, leaving only days. You have now seen how to perform single and multiple unit conversions. The key to performing these correctly is to 1) make sure you have all the conversion factors you need, 2) write out all of the steps and make sure the units cancel, and 3) think about your final result and ask whether the final result makes sense (is 30 minutes a small fraction of a day? Does 72 inches equal 6 feet?). 3 4 APPENDIX D: Uncertainties and Errors A very important concept in science is the idea of uncertainties and errors. Whenever measurements are made, they are never made absolutely perfectly. For example, when you measure your height, you probably measure it only to roughly the nearest tenth of an inch or so. No one says they are exactly 71.56789123 inches tall, for example, because they don't make the measurement this accurately. Similarly, if someone says they are 71 inches tall, we don't really know that they are exactly 71 inches tall; they may, for example, be 71.002 inches tall, but their measurement wasn't accurate enough to draw this distinction. In astronomy, since the objects we study are so far away, measurements can be very hard to make. As a result, the uncertainties of the measurements can be quite large. For example, astronomers are still trying to refine measurements of the distance to the nearest galaxy. At the current time, we think the distance is about 160,000 light years, but the uncertainty in this measurement is something like 20,000 light years, so the true distance may be as little as 140,000 light years or as much as 180,000 light years. When you do science, you have to always assess the errors on your measurements. 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21.
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