Chapter I Getting Your Bearings, Math Skills and The Sizes of Things

Chapter I Getting Your Bearings, Math Skills and The Sizes of Things Finding sizes: As part of our introduction to astronomy, you will be finding sizes of things and plotting the items. Part of the point is to get an idea of what is in the Universe. Part is to use scientific notation, review conversions and use the proper number of significant figures and use a logarithmic plot. You might need to find mass or radius or lifetime. The assignment is geared to Pathways to Astronomy, but you may use the internet to find the information. Be very careful to get what is requested. Download and save the Internet reference if you want to be able to show that the answer is correct. When using the book s index to find information, be sure to read all the words and to look on each page. You might find values that differ from other people s. For some objects, like the size of the Local Group, there is no firm outside edge and measurements of the size are all the same from one researcher to another. This does not mean that one value is right and the other is wrong. Converting Units: We compare features objects in order to get a concept of the Universe. The term size is somewhat vague. It might mean mass, radius, area, circumference. None of these is directly comparable to another. So we need to compare consistent measurements. Even when we have comparable measurements e.g. lengths for everything, the values all need to be in the same units so that we can understand what is larger. What if we don t have the information in the same units? It is necessary to convert them to the same units. For example, you might be seeking the size of an automobile. Possible answers would be the length of a Mini Cooper, about 14 inches long (.61x10 0 meters in scientific notation) and.4 inches (1.47 x10 0 meters) high. A Lincoln Navigator might be.x10 0 meters long and about 1.9 meters high. We converted the lengths or heights of the vehicles to the same units to see which is larger. On the other hand, the mass of the auto, or the surface area are different from one another and different from the length or height. They cannot be compared directly. Units- In science (and almost everywhere but the USA) use the metric system. So lengths should be in centimeters, meters, or kilometers. Time is usually in seconds. Mass would be in grams or kilograms. In astronomy there are some other unique units. These include Astronomical Unit distance between Earth and Sun 1AU=1.496x10 km=1.496x10 11 m Light Year(ly), distance light travels in a year Parsec(pc) is the distance of a object whose parallax is 1 second of arc 1pc=.6 light years 1 pc= 066 AU 1pc =.06x10 1 km 1pc =.06x10 16 m 1kpc=10 pc 1Mpc=10 6 pc 10 millimeter (mm)=1 meter 100 centimeters (cm)=1 meters (m 10 meters (m) = 1 kilometers (km) 1 mile (mi)= 1.609 km 1 mile=0 feet 1 meter = 9.7 inches 1 inch = 0.04 m (beware, meter is abbreviated m and mile is abbreviated mi) 1 foot = 1 inches Converting 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 Chapter I Getting Your Bearings, The Sizes of Things 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. 9.460x10 1 m 1 light year 1.496x10 11 m 1AU.07x10 16 m 1 pc 1km.97x10 4 in 1 in.04 m All of these values are equally true, but each is most useful when the units of the denominator (bottom of the fraction) are the same as the units of the value you want to convert, so that the units cancel. Read through the examples to see how it works. 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. 9.9 AU 9.9AU x 1.496 x 1011 m 1 AU 9.9 x1.496 x 10 11 m =1.47x10 1 m Cancel the AU to get Example: Sirius is.7 parsecs away from the Sun. How far away is this in meters?.07x10.7 pars ecs=.7 pc 1pc 16 m 16.19x10 m The pc cancel. Example: The distance to Vega is.9x10 17 m. How many light years is that? 1 light year Multiply the distance by, with the meters on the bottom to cancel the units. So 9.46x10 1 m 1 light year.9 10 17 m 9.46 10 1 m.9 1017 light year Since 9.46x10 1 was on the bottom of the fraction, we divide BY it to get 9.46 10 1.6 light years When you start, there is no equation. Just write the number and equate it to itself times ONE. You can tell that the conversion is correct if the units (just the names like pc or km) cancel top and bottom. Often, you will 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 only then to multiply all the values together. Example: A football field is 00 feet. How many kilometers is that? Find the conversion factors to go from the original units, step by step, to the units you want. 00feet 1inches 0.04m 1km 1foot 1inch 1000m = 600 inches 0.04m 1km 1inch 1000m = 91.44 1km 1000m = 0.09144km Check yourself: Convert each of the following a) 1.7x10 11 inches to m b) 14 yards to miles c) Convert 1 Mpc to m d) 14kpc to m e).4 ft to m Chapter I Getting Your Bearings, The Sizes of Things

a) 1.7x10 11 inches=4.1x10 9 m b) 14yd x ft x 1mi 7.9x10 mi 1yd 0ft c) 1Mpc x 106 pc 1Mpc x.09x1016 m d) 14 kpc = 1pc 4.017x10 m 4.6x100 m e).4 ft = 1.04x10 0 m CALCULATOR NOTE: The order of multiplication and division doesn t matter. On the other hand, many calculators need more parentheses than you might think. Example: If you type 6/x7 into a calculator, you get.4 not 0.171. The calculator will divide 6 by and then multiply the answer by 7. To divide 6 by x7, you need to type 6/(x7) or 6/=/7=. SCIENTIFIC NOTATION; This is a style of numbers. Scientific notation does not change the information, but does make it easier to compare values and eliminates the need to write out many zeroes. In Scientific Notation the value is written as a number between 1 and 9.9999 times an integer power of 10. So the following are written in scientific notation: x10 1, 4.x10-6, 1.09 x 10 4 while the following are not: 0, 4.9x10-7, 0.00109 x 10 7 although these numbers represent the same values. Your calculator will not care what format you use/ To put a number into scientific notation, move the decimal point until there is ONE non zero digit to its left. Count how many places you have moved it. Then multiply by 10 to the power equal to the number of places that you moved the decimal. If you moved the decimal to the left, making the number smaller, then multiply by 10 to a positive power. If you moved it to the right, making the number larger, then multiply by 10 to a negative power. If there is no decimal showing to start with, it is to the right of the last digit. For example 4.=.4x10 0.004=.4x10 - Plotting: We want to plot the sizes of astronomical objects to compare them. The wide range of sizes doesn t usually fit on a single plot, so we use a logarithmic plot, like the following. A logarithmic plot gives equal space to equal powers of 10, rather than giving equal space to equal size intervals. The major divisions start at 1 times a power of 10. Numbers that are not exactly powers of 10 (e.g. 4 x 10,. x10 etc) are plotted between the major divisions. The tiny numbers,,, show which mark to use for times the power of 10 that is immediately below, times, etc. The tic marks show all of the multipliers (x, x, 4x etc). They are not all labeled because there is too little space. To use the graph, express the value in scientific notation (a value between 1 and 9.999.. times an integer power of 10) in the correct units. Since the number is LARGER than the power of 10, its value will be plotted ABOVE the power of 10, at the position indicated by the leading number. For example, x10 9 m appears above the 1x10 9 line at the value. The value 1.x10 19 m appears slightly above the 1x10 19 line, but below the x10 19 line. When you plot the values you have found on the logarithmic graph provided and label them with the name of the object. The vertical coordinate on the plot is the size of the object in meters. The horizontal coordinate on this graph has no meaning (although it does on other graphs). Each value is plotted at or appears above the line with the power of 10. So value 1.x10 19 appears slightly above the 1x10 19 line, but below the x10 19 line. When you plot this on the logarithmic graph label the iem with the name of the object. Theory Note: How is spacing of the numbers decided? Every positive number can be represented by 10 raised to some power. This power is called the logarithm of the number. The power is not usually an integer. The spacing is determined by this power. So, for example, 4=10 log (4) =10.60, so the 4 is about 0.6 of the way from the bottom of the interval to the top. The spacing Chapter I Getting Your Bearings, The Sizes of Things

Sizes in Meters 1.0E+00 1.0x10 0 1.0x10 1.0E-01-1 1.0x10 1.0E-0-1.0x10 1.0E-0-1.0x10 1.0E-04-4 1.0x10 1.0E-0-1.0E-06 1.0x10-6 1.0E-07 1.0x10-7 1.0E-0 1.0x10-1.0E-09 1.0x10-9 1.0x10 1.0E-10 1.0x10 1.0E-11 1.0x10 1.0E-1 1.0E-1 1.0x10 1.0x10 1.0E-14 1.0x10 1.0E-1 1.0x10 1.0E+1 1 1.0x10 1.0E+14 14 1.0x10 1.0E+1 1 1.0x10 1.0E+1 1 1.0x10 1.0E+11 11 1.0x10 1.0E+10 10 1.0x10 1.0E+09 9 1.0x10 1.0E+0 1.0x10 1.0E+07 7 1.0x10 1.0E+06 6 1.0x10 1.0E+0 1.0x10 1.0E+04 4 1.0x10 1.0E+0 1.0x10 1.0E+0 1.0x10 1.0E+01 1 1.0x10 1.0E+00 0 1.0x10 1.0E+0 1.0x10 1.0E+9 1.0x10 1.0E+ 1.0x10 1.0E+7 7 1.0x10 1.0E+6 1.0x10 1.0E+ 1.0x10 1.0E+4 1.0x10 1.0E+ 1.0x10 1.0E+ 1.0x10 1.0E+1 1.0x10 1.0E+0 1.0x10 1.0E+19 19 1.0x10 1.0E+1 1 1.0x10 1.0E+17 17 1.0x10 1.0E+16 16 1.0x10 1.0E+1 1 Chapter I Getting Your Bearings, The Sizes of Things 4

Math Notes Order of Operations Do whatever is within parentheses first. Then: Raise to powers (i.e. evaluate the effect of the exponent) Multiply and divide (order doesn t matter) Add and Subtract (order doesn t matter between these) So, for example 6 + 9 +4x/17 =? would be evaluated as follows. Do powers first, then multiplication and division, then addition and subtraction. (6 ) + (9 )+{4x(/17)} or (6 )x(9 )+{(4x)/17} =16 + 1 +0/17= 97+0/17 =97+1.176=9.176 When using a calculator, be prepared to use parentheses in more places that you might expect. For example, the expression, 6x 7 should come out as.. The order of multiplication and division doesn t 4x matter, but the expression includes dividing by. A calculator may give (6x7/4)x = 1., depending on how the information is entered. The calculator needs to be told explicitly to divide by. Exponential Notation The symbols 10, 1E, and 10** all mean 10 to the third power i.e. 10 x 10 x 10. The first way of writing is used if the writer can do superscripts, the other ways are used in some computer notations where superscripts are not available More generally 10 n means 10 multiplied together n times, i.e. 10x10x10...x10 where there are n repetitions of 10 (or equivalently 1 with n zeroes following it.). To multiply together two numbers of the form 10 n x 10 m, the exponents are added, e.g. 10 n x 10 m =10 m+n Similarly, 14 means 14 x 14 x 14 x 14 x 14 = 74 not 14x10. A negative exponent, means 1 divided by the number raised to the power given. e.g. 10-1 means 1/10 - means 1/ =1/4 10 - means (1/10) multiplied together twice, i.e. 10 - =1/10 = (1/10) x (1/10) = 0.01 Similarly, 17 - = 1/(17 x 17 x 17) =.0x10-4 = 0.0000. The rule of adding the exponents when multiplying numbers of the form 10 -n x 10 m holds so 10 -n x 10 m =10 -n+m =10 m-n or subtract the exponent on the bottom when dividing. For example 10 m /10 n =10 m-n In math, two negatives cancel (or two wrongs make a right, if you like), so 10 m /10 -n =10 m (-n) =10 m +n Examples 10 9 / 10 1 = 10 9-1 = 10 10 9 /10-1 = 10 9-(-1) =10 1 Chapter I Getting Your Bearings, The Sizes of Things

Addition/Subtraction The calculator will take any combination, but there is no simple way to add exponential numbers, by hand. You need to either write them out explicitly or factor them e.g. 10 +10 4 = 100+10000=10100 =10 x (1+10 )=10 x (1+100)= 10 x (101)=10100 Computing with Scientific Notation and Similar Forms Because the order of multiplication and division doesn t matter, scientific notation can be used in computation as follows..1x10 6 x 7.x 10 9 =.1 x 7. x 10 6+9 =. x 10 1 =. x 10 16 in scientific notation. It is necessary to increase the exponent to compensate for decreasing the value. to. and increasing the exponent. It is NOT necessary to separate the powers of 10 from the rest of the computation. It is helpful if you are doing the computation by hand, because it is easy to compute the powers of 10 and estimate the rest. In the computation about, the x7 part could be estimated as 1. That would not be final answer, but it would give a rough idea of the result. Solving Equations Often we have a formula where we know values for some, but not all, of the variables. For example Distance =Rate xtime Distance= 40 kilometers, Time= hours The goal is to rearrange the values so that an unknown value appears on one side, alone, and only known values appear on the other. The way to do this without destroying the information embodied in the equation is to use EQUAL TREATMENT for both sides of the equation You can do the following without making an equation untrue: Substitute a numerical value for a variable, if you know the value Multiply or Divide by something, both sides (using zero here is legal, but it destroys the process) Add or Subtract something, both sides Raise something to a power (like square it, take the square root) Clear parentheses, for example a(b+c) = ab + bc and you should do one or another of the above (or several) if it helps to get all the known things on one side, and the unknown things on the other. Looking back at the problem, Distance =Rate xtime Distance = 40 kilometers, Time= hours We know the distance and the time and want to find the rate. So divide both sides by Time, to get rate by itself (1/Rate) x Distance =Rate x Time x (1/Rate) The point was to be able to cancel the Rate, so we do to get (1/Rate) x Distance =Rate x Time x (1/Rate) Distance/ Time = Rate Now it is easy to substitute, (Distance = 40 kilometers, Time= hours) 40 km/ hr = Rate 11. km/ hr = Rate Chapter I Getting Your Bearings, The Sizes of Things 6

Algebra Advice Solutions in algebra are not very intuitive. Don t expect to know the answer automatically, no matter how brilliant you are. It is rather like fixing a car or cooking. No matter how smart you are, it still takes all the steps and a bunch of time. It doesn t mean that you are dumb or bad at math. It just works slowly and systematically. It is best to write out EVERY step completely. Be sure that each line is a repetition of a true equation. That way, you can understand what you have done if you come back. Not every legal thing you may do will help isolate the unknown. If it doesn t help, it may be necessary to start again. Even if it seems to have helped, it may be worthwhile to put the problem away. A good way to check your work is to put the answer aside, separate from the problem. Come back to it when you have not thought about it for a while. Redo the problem. NOW compare your results. If they are different, at least one must be incorrect. If you are working with a friend, compare methods, each reading the other person s. Or read the work out loud to someone, even if they have not done the problem. It will make you go over the information slowly and often help you to notice an error. It is hard to read your own work and find the error. We all want to be right and that gets in our way as we look at our own results. UNCERTAINTY Most numbers we use are the results of measurements. So they are never perfectly accurate. Unfortunately not all textbooks tell you how accurate the values are and it is not true that the last digit is the only one that is uncertain, Consider a value like.x10-9. The range of values really might really be from 4.0 x10-9 to.x10-9 with probability 9% (typically the exponent would not change). Another way that this can be written is. x10-9 ( +0.1, -0., ). There is no way for you to know the uncertainty is unless the author specifies it. In this case, we would say that the value has 1 significant figure, since the range of values from largest to smallest leaves only one digit unchanged. SIGNIFICANT FIGURES Another indication of the precision of the information is the number of significant figures in the number. If you have a number in scientific notation, the number of significant figures is the total number of digits in the coefficient. (It isn t necessary to write a value in scientific notation to figure out the number of significant figures. But it does clarify.) The power of 10 doesn t count. Zeroes on the right do count. Don t ignore them, put on more or take them off. If your calculator adds them, remove them. (There won t be any leading zeroes on the left in scientific notation.) Examples: 0.09 has 1 significant figure (9. x 10 - ) 0.010 has significant figures since the 0 following the 1 indicates that you know the next digit. You could write it as 1.0 x 10 -. The leading zero(s) indicate the power of 10, but is (are) not considered to be significant figures 9.7 x 10 9 has significant figures How many significant figures should be used? If a computer program, like Desire Learn specifies the number of significant figures, do what it says. The number of significant figures in the answer to a multiplication or division problem (or any combination) is the same number of figures as in the LEAST precise of the values included in the computation. Use all the digits you have for your calculation and round off to the number of figures at the END. Examples 9.7x10 9 /.449x10 1 =.6 x10 - Keep only significant figures since 9.7 has only. 0.09 x 0.010/9.7x10 9 = 9x10-14 should have ONLY 1 significant figure, the same as the least accurate value in the computation. Chapter I Getting Your Bearings, The Sizes of Things 7

The number of significant figures to keep from an addition or subtraction is tougher. Write the numbers out with the 10 s place, 1 s place, 0.1 s place etc lined up. Keep only figures that come from columns where every number has a specified digit. Trailing zeroes DO matter. So, for example.4 +6.76 =10. We cannot write 10.16 because we don t know what added to.06. 7.9x10 +.1x10 9 = 7.9 x10 +1. x10 =.4 x 10 =.4 x 10 9 The from 7.9 doesn t show up because there is no digit in the same column. If you use your calculator, you get.94x10 9. It is up to you to round to the proper number of significant figures. If addition or subtraction is involved in a computation, the numbers need to be written out, and digits should be kept so long as they are the result of known values. Example.4 x 10 +04+6.6= Adding 4? ( actually we don t know the 0, but it is needed to keep the columns straight) 04 6.6 06.6 This value does not include the correct number of significant figures. If we write it in scientific notation the result becomes.066x10, with all the digits, and.1 x10 with the correct number of digits. The two 6 s resulted from addition of some uncertain values. The final answer was rounded up to get the best value When using a formula, like R, the numbers and are exact. So the has infinitely many significant figures. The value of has as many digits as you write, or as the calculator has (like 9 or 11), so that is not usually an issue. Conversion Factors 1 meter = 9.7 in (meter is abbreviated m) 1 foot = 1 in 1 in =.4 cm 1 mile = 1.609 km 1 statute mile=0 feet (this is the normal kind of mile, it would be abbreviate mi, NOT m) 1 nautical mile= 600 feet 1 nautical mile= 1. km 1 m = 100 cm 1 km = 1000 m 1 = 10 6 m ( the symbol, like m in the Greek alphabet, stands for micron unit) 1 nm = 10-9 m 1 Å = 10 cm=10-10 m (the symbol Å stands for Ångstrom units) 1 AU = 1.496x10 km (AU stands for Astronomical Unit, the average distance from the Earth to the Sun) 1 ly = 9.46x10 1 km (ly stands for light year, the distance that light would travel in a year) 1 pc =.6 ly Chapter I Getting Your Bearings, The Sizes of Things

1 pc =.07 x10 1 km (pc stands for parsec, the distance at which parallax shift due to the observer moving by 1 AU is one second of arc) 1 kpc = 10 pc 1 Mpc = 10 6 pc Triangles If you know two sides and one angle, you can finish drawing the triangle. Which means, the triangle is known. e.g. Two Sides and The Included Angle a C b Two Sides and Another Angle a A b If you know all three sides, you can also finish the triangle, i.e. draw and stick it together in only one way. So you know the entire triangle On the other hand, if you know all the angles, you know the shape of the triangle, but not its size. Triangles with the same shape as one another are called similar, like the following three triangles. a c b The angles in any plane triangle always add up to 10 degrees. Angles 60 o in a circle, 10 o in the angles of a triangle or on one side of a straight line 1 o =1 degree=60' ( minutes of arc) 1'=60" (seconds of arc) Areas Triangle Area = 1/ Base x Height Rectangle Area = Base x Height Circle Area = Radius, where =.1419... The circumference of a circle is Circumference = Radius. The number of degrees in a circle is 60. Trigonometry, just so you know A special, useful class of triangles is the right triangle. Since one of the angles in it is always 90 degrees, the right angle, and there are 10 degrees total, there are 90 degrees left among the other two angles. Once one of the angles is known, the other is simply the remainder of the 10 degrees. Trigonometric Functions Hypotenuse o 90-A sine A = Opposite Opposite Hypotenuse 90 o A The right angle cosine A = Adjacent Hypotenuse Adjacent Opposite tangent A = Adjacent Since the shape of a right triangle is known once one of the smaller angles is known, we can understand the relations between the lengths of the sides. The relations are tabulated as the functions shown above. The ratios of the lengths of the sides can be found in books or on scientific calculators. Be sure that the calculator is in degrees mode to use it to find a trigonometric function. Chapter I Getting Your Bearings, The Sizes of Things 9

One more quick trick. If an angle is smaller than about 10-1 o, then sine or tangent of the angle is about the same as the size of the angle in radians. Angles 60 o in a circle, 10 o in the angles of a triangle or on one side of a straight line 1 o =1 degree=60' ( minutes of arc) 1'=60" (seconds of arc) 1 radian = 7.9... o radians =60 o Areas Triangle Area = 1/ Base x Height Rectangle Area = Base x Height Circle Area = Radius, where =.1419... Sphere Area = 4 Radius The circumference of a circle is Circumference = Radius. Chapter I Getting Your Bearings, The Sizes of Things 10