Proton. Size of cell. 100 = 10 2, so the logarithm of 100 is 2, written Log 100= 2

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1 Homework 1 Date Due Name You will be making a chart of the sizes of things in the Universe. It should come out similar to Figure., but more precise. The plot you will be working on is at the end of this hw. This assignment should give you a picture of what is present in the Universe, the relative the sizes of things, ensure that you learn to do conversions, and learn how a logarithmic plot works. What is a logarithmic plot? Let s first look at the plot in Figure.. The vertical scale shows the size of the object (length, width, height, diameter etc) in meters. It is labeled at intervals of m. Although there are no intermediate markings, things can, of course, have intermediate sizes. Each power of 10 takes up the same amount of space. This type of plot is called Logarithmic. A logarithm is the power of 10 that could be used to represent a number. For example 100 = 10, so the logarithm of 100 is, written Log 100= 1000 = 10 the logarithm of 1000 is, written Log 1000= Similarly 1 = 10 0 Log 1 = 0 0.1=10-1 Log 0.1 = -1 Other positive numbers can also be expressed as powers of 10. So = Log = = Log = = Log = =x10 1 = Log 0 = In the logarithmic plot, the spacing of the numbers is based on the spacing of the logarithms (powers of 10) although the numbers themselves are on the scale. Your scientific calculator has a key that finds logarithms. Look for Log, probably on the same key with 10 x or 10 y. Try entering 100 and see that you get as its log. Why use a Logarithmic plot? The sizes of things in the Universe vary enormously, so the Logarithmic plot allows us to fit everything on one page. In some cases relationships show up better when the logarithm is plotted. In Figure., the horizontal axis is not used, so it is like a number line or a ruler. Reading the logarithmic plot A plot is useless if we are unable to read information from it. What is plotted on figure.? As you can see, the marks are equally spaced at intervals where the size is multiplied by That is a very large factor and there isn t a lot of space. So it is not possible to read or plot on this diagram very precisely. (The other columns are mass and time. The columns are not related ) Consider the a kiloparsec. A kiloparsec is 1000 parsecs and 1 parsec is.09x10 16 m (see inside front cover or appendix table I). So 1 kiloparsec=1000 x 1pc=1000x.09x10 16 m=.09x10 19 m. It is plotted a little lower than10 0 meters. Similarly, look at the Height of human. Most adults are between one and two meters (9.7 to 7.74 inches). So the mark is a little above 10 0 m. Object Practice Reading a Logarithmic Plot Read off and record the sizes of the from Atom figure.. Use scientific notation for your Proton answers (a number with one digit before the decimal point, then a power of 10). Size of cell As you can see, it is hard to find the exact correct value. (yes, turn this in). Size in meters Astronomy HW 1 1

2 A logarithmic plot can be more precise, like the one at the end of this hw. In this case, all the columns are part of one plot. The columns are to make it fit on the page. The bottom of the middle column is the same as the top of the left column etc. An expanded version of part of the plot is shown. One times each power of 10 is labeled. The numbers above the power of 10 are x that power, x that power etc. There is no Zero on this plot and there are no negative numbers. The small numbers indicate how many times the power of 10 just below. Not all the values are marked because of the small space available. To plot a number, change it to scientific notation (a number between 1 and times a power of 10) and then plot it above the power of 10. Examples 106. =1.06x10 Move the decimal point to the left three places, so there is 1 digit in front of the decimal. Then multiply by 10 to keep the meaning the same. Plot above 1 x 10, between 1 and as is shown on the next page =.x10-4 Move the decimal point to the right four places, so there is 1 digit in front of the decimal. Then multiply by 10-4 to keep the meaning the same. This number is MORE than 1 x 10-4 so it goes above 1 x 10-4 as shown on the next page x 10 6 =1.700 x10 11 Move the decimal place to the left five times as in the first example, but ALSO combine the powers of 10. Plot above as shown on the next page. Height of a Human estimated at 1.6 meters (ft in) =1.6x10 0 m See plot. WHAT TO DO A. Look up the sizes requested in the table on the next page. Record each size (diameter, length, width or height) with whatever units you found in the first column. B. Change each value to meters and to scientific notation and record the value in the right hand column. If the value you found was already in the correct form, don t change anything. You are not required to show your work. On the other hand, if there is an error, you may get partial credit if you show work. Just write things down on a piece of paper and attach it to whatever you turn in. Keep at least digits after the decimal if you are doing a computation. C. Plot each value on the logarithmic plot. Draw an arrow to show exactly where you want the number and LABEL each value with the NAME of the quantity. Remember the values are at least 1 x power of 10, so they go AT or ABOVE the power of 10. D. Fill in the table on p 1 Ans to examples a. 6.47km b 9.7x10 m c..04x10 m d x10 0 Astronomy HW 1

3 Quantities to Plot Sizes of some of these items can be found in Unit. The rest can be found in elsewhere in Pathways. Look in the Index, Appendix or Glossary. Be sure to read all of the referenced pages and possibly a little more. Look at the figures as well as the words. If you use a source for your information other than Pathways, be prepared to send or show me the information if I challenge the value. If the source is basically sensible and says the wrong number, you will get credit. Object Earth Moon Moon s Orbit Sun Earth s Orbit (semimajor axis, a, of Earth s orbit) Solar System to Neptune (semimajor axis, a, of Neptune s orbit) Distance to Nearest star (Proxima Centauri, don t double it) Milky Way Galaxy Local Group Local Supercluster Visible Universe Ceres (an asteroid) Kuiper Belt White Dwarf Canis Major Dwarf Galaxy (distance to it) Comet (including tail) Diameter, or Length or Width or Height in whatever units you find. Be sure to specify units. Diameter, or Length or Width or Height in Meters in Scientific Notation Calculator notes: Using powers of 10. Scientific calculators all have a way to express scientific notation. Look for a key labeled EXP or EE or ee or E. These may be in the second or inverse function position. These keys all mean, what follows is x 10 to the power typed. So 10 is written 1. EE. The screen on your calculator might show 1. E 0 or 1. x 10 or it might read 1. 0, the raised is the exponent. DO use the EXP or EE or ee or E key, to ensure that the calculator uses the number as all one thing. DON T type 10, when entering an exponent. If you type 10 the number will be 10 times what you want. You should enter 10 4 as 1 EXP 4. When reading out an answer, be sure to Astronomy HW 1

4 include x 10 and the exponent. Don t use a multiply sign inside a single number. The calculator will divide wrong. The calculator doesn t care if you do a calculation with some numbers in scientific notation and others not. Parentheses The order of multiplication and division doesn t matter. On the other hand, many calculators will need more parentheses than you might think. Example: If you type 6/x7 into a calculator, you get.4 (wrong) not (correct). The calculator will divide 6 by and then multiply the answer by 7. To divide 6 by x7, you need to input 6/(x7) or 6/=/7=. Conversions When values are to be compared or added, they must be in the same units. Converting units does not change their meaning. But, as you know from algebra class, the only things that can be done to a number without changing its value are to add 0 or to multiply by 1. To convert multiply by 1. This may sound useless, how can multiplying by 1 do anything at all? To convert units, you might use any of the following. They are all equal to 1 since they are the same on the top and bottom. All of these values are equally true, but each is most useful when the units of the denominator are the same as the units of the value you want to convert Example: Saturn s orbit has a semimajor axis of 9.9 AU. How large is the semimajor axis in meters? To get from AU to meters, look up 1 AU = 1.496x10 11 m Since these values are equal, they can be placed, one over the other, to make a form of 1. How should it be done? The units of the previous value (the AU) should be on the bottom, to cancel. Please notice that m stands for meters, not miles. Miles would be abbreviated as mi. Example: Sirius is.7 parsecs away (basically from the Sun). How far away is this in meters? When you start, there is no equation. You just write the number and equate it to itself times ONE. But be sure to express ONE using two values that mean the same, but are in different units. You can tell that the conversion is correct if the units (just the names like parsec or km) cancel. You may not find a single equation relating the original units to the final value. In that case, find relations between the current units and some other, then between that unit and another, etc. etc. until you have steps relating one unit to the next, without skipping any steps. In this case you will be multiplying by 1 several times. It is generally better to write out ALL the terms, cancel the units (to be certain that the correct values are being used) and then multiply all the values together. Example: Your height is 1. yards. Express it in m, in scientific notation.. Check yourself: WebCT has a practice quiz to use to practice conversions. It has lots of variations. For paper practice convert each of the following a) 4 miles to km b) Mpc to meters c) 1.x10 7 inches to m d) 4. kpc to m Astronomy HW 1 4

5 Sizes in Meters 1.0x x x10-1.0x10-1.0x x10-1.0x x x10-1.0x x x x x x x10-1.x10-4.x x x x x x x x x10 1.0x x x10 1.0x x10 1.0x10 1.0x x x x10 1.0x x x10 1.0x x x10 1.0x x10 1.0x10 1.0x x x x x x x10 0 m Height of a Human 1.0x10 1 Astronomy HW 1