Review of Mathematical and Scientific Methods

Transcription

1 NAME: Grade: / = % Review of Mathematical and Scientific Methods Objectives To review the essentials of mathematics: algebra scientific notation exponentiation (powers of 10) logarithms significant figures To review the essentials of math and the scientific process as they relate to astronomy: small-angle formula scaling uncertainties in measurements statistics scientific method Materials 15-cm Ruler (we measure using the metric system) Scientific Calculator (borrow one if necessary) Background and Theory Mathematics is the language of the Universe. Imagine learning English and not being able to use any vowels: or given a coded message without a key: Similarly, astronomy cannot be taught adequately without using the proper language. Astronomy 101 covers the basic concepts in astronomy and astrophysics (really the same thing). Some concepts can be explained best and most completely by the use of a formula. You may be surprised to see how much information is contained in a simple algebraic equation; for example, Einstein's famous equation: E = mc 2.The math involved is at the level needed to be accepted into this great university. If you happen to be a 'mathophobe' (or arithmetically challenged), please do not worry about the math used in this course; you will always have your fellow students and your instructor to help. You will be well prepared for any exam questions involving math. Procedure Each of the following steps of your review starts with a sample problem -- one that is exactly like or similar to a problem you will be seeing or have seen during this semester. Work the problem given. If you cannot work the problem quickly, without hesitation, then refer to one of the review sheets, ask one of your partners, and finally ask us for help. Students arrive at universities with a broad range of math backgrounds. It may be that you do not need much of a math review; use your talents by helping your classmates through this practice. Note, however, that the scientific methods section includes applications specific to astronomy.

2 Review of Mathematics 1.1. Algebra (2 points) A. Solve for y: B. Solve for m (mass): Understanding: In the equation above, E represents the energy; m, the mass; and c the speed of light -- approximately 300,000 km/sec. Why can a whole lot of energy be obtained from a tiny bit of mass? Answer: Because the energy is the product of the mass and a very large number. The speed of light, squared, is 90,000,000,000 (km/s) Scientific Notation (2.5 points, 0.5 points each) Write the following in scientific notation: A. 3,042 B A Convert the following numbers from scientific notation: B. 4.2 x 10 9 A. 4 x Exponentiation (powers of 10) Work the following problems using a scientific calculator. If you do not own a calculator that uses scientific notation, then find someone who does and borrow it, or get to know them. (3 points total; 1 point each) B. Multiply: 3.1 x 10 7 by 3 x 10 5 A. Simplify: (3 x 10 8 ) 2 B. Simplify: 1.4. Logarithms (3 points) When working with powers of 10, we will also need to reverse the process and calculate the log of a number. The log of a number tells us the equivalent number that we raise 10 to. For example: the log of 1000 is 3 because 10 3 is The log of 6000 is approximately because is (about) We would have guessed that the log would be between 3 and 4 because 10 3 is 1000 and 10 4 is 10,000, and 6000 lies between those two numbers. How does one calculate the log of a number? By using a calculator. Note we will not be using the natural log, or ln. Find the log button on your calculator; also, find the 10 x button to reverse the process.

3 Solve for M in the equation M = m 5 log (d) + 5 when: m = d = M = (A) (B) (A) Significant Figures The guidelines for significant digits are: Carry one or two non-significant digits through all calculations. Round the final answer to the required number of significant digits. The number of significant digits will be that of the value having the smallest number of significant digits How many significant digits are in the following numbers? (2 points total, 0.5 points each) B 1.5 A B 4000 A Multiply 1.5, , 4000, and Round your answer to the correct number of significant digits. (2 points) (A and B) 2. Astronomical and Scientific Methods 2.1. Small angle formula For small angles (much, much less than 1 degree), the sine of the angle is approximately equal to the tangent of the angle, which is approximately equal to the value of the angle itself. (Find your calculator and check this fact out. Your angle should be stated in radians.) If we know the distance to an object and can measure its angular size, we know its actual size. If we know the actual size and can measure the angular diameter, we can determine its distance. (note: This also works if the angular size is given in arc seconds, as most measurements are in astronomy since objects such as stars and galaxies are SO far, far away. The angular size is very small!)

4 2.2. Practice: (3 points) Angular size in radians Actual size in kilometers Distance in kilometers (B) 0.01 (diameter) 3480 (diameter) (A) (diameter) 150,000,000 (B) 6800 (diameter) 5.5 x 10 7 We observe 5 Ferrari Testerossas and note that they appear to be different sizes. We automatically know since they are the same model that the cars must be at different distances. Even if they are different colors, their morphology (their structure, how they look) tells us that they are the same kind of car. Because astronomical objects are so far away, we have no depth perception to help us determine their distances. We need to rely on other methods. Note in the above sketch that the Ferrari's are all the same actual size, but that the angular sizes (shown by the triangles to the left) are decreasing with distance. If the distances are great enough, then the angle we measure is directly proportional to the actual size of the car and its distance: Since the size of the object is constant, if the angle appears to be 1/2 as great, then that car must be 2 times farther away. We can always judge the relative distances this way; to obtain actual distances, we must have a reliable measurement of the closest car (in this example). Here s an example of what we want to avoid: Are we dealing with Mini-me or Dr. Evil? Which one is closer? Farther? What if they are at the same distance? How would you know?

5 2.3. Scaling and Scale Factors If you've ever read a map, used the scale to determine how many miles were represented by an inch, and then used a measured number of inches to determine how far you were going to travel, then you have all the skills needed to start your career as an astronomer! Think about how the cartographers calculated the map scale in this case: someone had to measure the actual number of meters or yards on the ground and then, perhaps using a ruler, translate that to inches on the map. (A) How far, in kilometers, is it from my office at the Institute for Astronomy (red dot) to our lab here in PSB (blue dot)? (1 point) (B) You live off campus at the corner of Oahu & Kamehameha (green dot). You are late to lab in PSB, so you ride your bike. You are very safe and take only streets. Draw your path on the map. How far is it from home to PSB (in meters) on streets? (2 points)

6 (A) You ride quickly, at 5 meters per second. How long does it take you to get to lab? (1 point) 2.4. Measurement Errors and Uncertainties (A and B) The term "error" signifies a deviation of the result from some "true" value. Often we cannot know what the true value is, and we can determine only estimates of the errors inherent in the experiment. If we repeat the experiment, the results may differ from those of the first attempt. We can express this difference as a discrepancy between the two results. The fact that a discrepancy arises is due to the fact that we can determine our results only within a given uncertainty or error. The more precise our measuring tool, the more precise our answer; the more accurate our input data, the more accurate our results. But, we will always have some uncertainty. In science, we can never hope to have absolute precision, absolute accuracy. (Thought question: What is the difference between a "precise" measurement and an "accurate" measurement?) It is perfectly acceptable to have large uncertainties, as long as we let the reader know our estimate of those uncertainties so that an intelligent evaluation of the results can be made. As scientists, we strive to perfect our experiments and observations in order to improve the precision and accuracy of our results. Measure these white lines and write the answers below the lines: Line A Line B Line C (4 points) Which measurement will be the most precise? Least? Which measurement will warrant the greatest number of significant figures? Least? For which line could we justify the use of a micrometer capable of measuring to 4 decimal places? What are your estimates of your uncertainties (the possible errors in your measurements) for each line? NOTE: when determining your uncertainties, you are free to put down whatever number you feel is approximately how right you think you are (2 points) The measurement of the height of line C would be even more uncertain. Do you see why? Explain.

7 2.5. Statistics (A and B) We use an absolute minimum of statistics in this class. You are probably very familiar with the terms "mean" and "standard deviation" from your high school math as well as from large lecture classes here where the grades are based on a curve. The mean is the average, while the standard deviation gives a measure as to how spread out the data are. If we have enough data, we may wish to graph the observations and see if there is a relationship between the variables. Here's a trivial example. Assume that the Ferrari's mentioned above can go from 0 to 60 mph (about 100 km/hr) in 6 seconds, and that the acceleration is constant. Here is what the graph would look like: (3 points) What is the slope of the line (the change in y divided by the corresponding change in x)?. How fast would the car be traveling after 3.5 seconds?

8 Make sure you understand the basic thought processes you are going through to answer these questions! (3 points) Let's assume one of the other Ferrari's is having air intake problems. When we plot the data, we notice that the data do not fall on a straight line. We do not want to simply "connect-thedots" as we do not care about the second-by-second acceleration, just the overall relationship. In this case, one draws the best-fit straight line, attempting to get a balance between the number of data points above and below the line: What is the slope of the line?. How fast would the car be traveling after 3.5 seconds? What is the uncertainty in the slope of the line?

9 Angular sizes (A and B) Small angle formula: Astronomers may assume that if they see two morphologically similar galaxies, that these galaxies are similar in actual size. Therefore, if one of the galaxies appears to be one-half the angular size of the other, then the "smaller" galaxy is twice as far from us. Measure the diameters of the above three, similar (nearly identical) galaxies and express your answers in mm or cm. [For the REAL images, we would know the actual angular sizes and could relate the ruler measurements on our paper to the actual angular sizes in the sky. Here we will assume that their measured sizes are directly proportional to their angular sizes.] Diameters in mm or cm: (1.5 points) The angular size of a galaxy is inversely proportional to its distance from us. So, if a galaxy appears to be one-half the diameter of another galaxy, then the smaller galaxy must be twice as far away. How much farther away from us is the farthest galaxy compared to the nearest? (2 points) Now, how would we ever know if we were comparing Dr. Evils to Dr. Evils or Dr. Evils to Mini-Me s? How could the galaxy in the left-hand image be at the same distance as the galaxy in the middle and yet look so much larger (have a much larger angular size)? (2 points)

10 Scaling and Scale Factors (3 points) Figure out the approximate wavelengths (in nm or 10-9 meters) of the blue, yellow, and red lines in the above figure by the eyeballing-it method. Write your estimates here: Determine a rough "scale factor" or "plate scale. Measure the distances in mm or cm between each of the marked wavelengths and average your results in the table below. (4 points) Wavelengths (nm) Difference (nm) 400 & B 400 & A 600 & B 400 & A Distance (mm or cm) Average: Scale (nm/mm or nm/cm) Now, use a ruler to measure the distances between the vertical line representing 400 nm and the blue, yellow, and green lines. Write those measurements (in cm or mm) down and complete the table (6 points): Distance from 400 nm line to: Multiply Column 1 by your average scale factor More precise determination of the wavelength of the: blue line B blue line yellow line A yellow line green line B green line Working with Units and Ratios Units and ratios are important - units because we want to know if we are talking about right next door or across the Universe, and ratios because we want to avoid having to manipulate exponents and because the constants we use in astronomy cancel out and we can get to comparisons that are easier to understand. Bonus questions: (A and B) (2 points) The distance to Sirius is 8.6. The radius of the Universe is 15,000,000,000. What units are we talking about here? Kilometers? Astronomical Units? Parsecs? Light years? Megaparsecs? (2 points) The mass of Jupiter is approximately 1.9 x kg. The mass of the planet orbiting the star 51 Pegasi is approximately 8 x kg. How do these two masses compare quantitatively? Isn t it easier to understand the comparison if one says that the planet orbiting 51 Pegasi is about 45% of the mass of Jupiter?