ASTR 112 Lecture 2 - Dirk Grupe PDF

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Dirk Grupe...

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A few Words regarding Math • First thing: Do Not Panic • This is not rocket science (although we do this here too) • You can do it! • Math is our language in Physics/Astronomy to describe the world • All math used in this class is simple Astronomy vs. Astrology Solving problems • Analyze the problem • What is the important information? • Critical Thinking! • Use your toolbox • Apply to right tools! Solving problems • In Astrophysics we use math to express our laws • We express our observations with equations • Those are our tools! • Your brain is your tool box • Apply it! • Use the correct equations!! • Do not just blindly put numbers into a formula! • Math is your friend not your enemy!!! • Double check you results • Look at the units!!!!!! Solving problems • The math we will use in this class is simple! • Only algebraic expressions • Simple stuff that you use every day • Math will give you a better understanding about the underlying physics Example 1 When 3 bananas cost 90 cents in the store, how much do you have to pay for 5? Example 1 So, first thing is we have to figure out how much one banana cost: 90/3 = 30 cents Now we know that we want to buy 5 bananas, so the amount we have to pay is 5 time 30 cents = $1.5 Example 2 You drive in your red Corvette (build in Kentucky!), that you just bought for $60000 with a velocity of 100 km/h on I-64 between Morehead and Lexington which have a distance of 100km. How long does it take you to get from Morehead to Lexington? Example 2 • What is the important information here? • You drive in your red Corvette, that you just bought for $60000 with a velocity v of 100 km/h on I-64 between Morehead and Lexington which have a distance d of 100km. How long does it take you to get from Morehead to Lexington? • What is the relation between velocity, distance and time t? • Velocity is distance/time v= d t Example 2 • You drive in your red Corvette, that you just bought for $60000 with a velocity of 100 km/h on I-64 between Morehead and Lexington which have a distance of 100km. How long does it take you to get from Morehead to Lexington? • Velocity is distance/time • So then 100 km/h = 100 km/time • So to get the time we need to rearrange the equation to • Time = distance/velocity • In our case: • Time = 100 km / 100 km/h = 1 h • So it takes us one hour to drive from Morehead to Lexington Example 3 • An ICE-3 is travelling between Munich and Augsburg (Germany) with a mean speed of 160 km/h. It takes the train half an hour(0.5 hours) to travel between the two cities. What is the distance between Munich and Augsburg? v= d t Example 3 • So, what is the problem here? • Velocity is defined by distance/time: v = d/t • Then distance is velocity times time: d = v x t • What’s the important information: • An ICE-3 is travelling between Munich and Augsburg (Germany) with a mean speed of v = 160 km/h. It takes the train half an hour (t = 0.5 hours) to travel between the two cities. What is the distance between Munich and Augsburg? • So, this results in: d=v×t=160km/h×0.5h =80km

Formulae • Mathematics and formulae are our way to describe the world around us • Formulae describe physical laws • The letters and symbols represent properties • E.g.: • Where M is the mass of a central object • v is the velocity of a body around this mass • r is the distant of the body to this mass • And G is the gravitational constant G = 6.67 x 10-11 m3 s-2 kg-1 M = v2r G Formulae • Another example is Einstein’s famous E =mc2 • This relation relates mass to energy • where E is the energy, m is the mass and c is the speed of light (3x108 m s-1) • Or the relation between wavelength and frequency with is equal to the speed of light: c = λν • Where λ is the wavelength and ν is the frequency Very important: Stick to the same units!!! • When you deal with formulae it is important that you stick to the same units!!! • For example, let’s use the relation between wavelengths and frequency: • What is the frequency of red optical light which has a wavelength of 600 nm? • So if we just blindly plug in numbers we get garbage if the speed of light is given in units of m s-1 • We first need to convert the 600 nm into m • How much is 1 nm? Very important: Stick to the same units!!! • 1 nm = 10-9 m • So 600 nm are 600 x 10-9m or 6.0 x 102 x 10-9 = 6.0 x 10-7 m • With this we can now solve the question: • And we can now fill in the numbers: • So the frequency is 5 x 1014 Hz (Hertz). c=λ ×ν ⇒ν = c λ ν= c λ =3×108m s−1 6×10−7m =0.5× 108 10−7 s−1 =0.5×108−(−7)Hz=0.5×1015Hz Even More Scientific Terms – Metric System • Astronomers use the metric system (masses measured in grams or kilograms, distances measured in meters or kilometers) • Units are based on factors of ten = easier to use! Simplicity! • Compare 1 meter = 100 centimeters = 1000 millimeters to 1 mile = 5280 feet = 63360 inches Physicist use SI Units • SI = Systeme International d’Unites (= International system of units • Founded in 1875 as the meter convetion • 7 basic units: 1. Length: Meter (m) 2. Time: Second (s) 3. Mass: kilogram (kg) 4. Electric Current: Ampere (A) 5. Temperature: Kelvin (K) 6. Amount of substance: Mole (mol) 7. Luminous brightness: Candela (cd) Physicist use SI Units • All units can be based on natural constants • The advantage is not only that all units are based on the base 10 (not like your feet and pounds etc) .. • .. but other units can be derived from these 7 basic units • Examples: • Force: Newton: 1 N = 1 kg m s-2 • Energy: Joule: 1J = 1 N x m = 1 kg m2 s-2 • Power/Luminosity: Watt: 1 W = 1 J/s = 1 kg m2 s-3 • Pressure: Pascal: 1 Pa = 1 N m-2 = 1 kg m-1 s-2 • Frequency: Hertz: 1 Hz = s-1 Also important: Double check your results! • Make a check if the result you get makes any sense • For example if your calculations show that the temperature of the sun is 10-6 K, the obviously something is wrong (or we would be all dead) • Of if you calculations results in that it takes an aircraft 18 hours to get

of the runway, the you certainly do not want to be on that plane J • Learn what the symbols mean and go through examples. Numbers in Astronomy • To express numbers in astronomy in a manageable way, astronomers use Scientific Notation. – Mass of the sun: Either write as 2,000,000,000,000,000,000,000,000,000,000 kg or as 21030 kg – Mass of a hydrogen atom: Either write as 0.0000000000000000000000167 kg or as 1.6710-27 kg • Note the use of a base (2, 1.67) and an exponent (30, -24) in these two examples Numbers in Astronomy Some more examples • 0.001 à 10-3 • 0.01 à 10-2 • 0.1 à 10-1 • 1 à 100 • 10 à 101 • 100 à 102 • 1000 à 103 • 100000 à 105 • 1000000 à 106 Numbers in Astronomy • In order to make numbers more workable, Astronomers use logarithm to the base 10 – log10 or short log • The logarithm of a number is the exponent to which the base, here 10, has to be raise to produce that number. • For example the logarithm of 1000 is 3, • because 103 = 10 x 10 x 10 = 1000 • In general: • Don’t freak out (yet) • Logarithm make you life easier y=10x ⇒x=log(y) Numbers in Astronomy • When using logarithms, things can be reduced to simple additions and subtraction, instead of multiplications and divisions • General rules for logarithms: • For example: log(a×b)=log(a)+log(b) log a b " # $ % & '=log(a)−log(b) log(100×1000)=log(100)+log(1000)=2+3=5 log 1000 10 " # $ % & '=log(1000)−log(10)=3−1=2 List of some useful logarithms • 1 = 100 à log (1) = 0 • 2 = 100.3 à log(2) = 0.3 • 3 = 100.477 à log(3) = 0.477 • 4 = 2 x 2 = 22 à log(2) + log(2) = 2xlog(2) = 0.3 + 0.3 = 2 x 0.3 = 0.6 • 5 = 10/2 à log(10) – log(2) = 1 – 0.3 = 0.7 • 6 = 2 x 3 à log(2) + log(3) = 0.3 + 0.477 = 0.777 • 7 = 100.845 à log(7) = 0.845 • 8 = 2 x 2 x 2 = 23 à 3 x log(2) = 3 x 0.3 = 0.9 • 9 = 3 x 3 = 32 à log(3) + log(3) = 2 x log(3) = 0.945 • 10 = 101 à log(10) = 1 • 20 = 2 x 10 à log(2) + log(10) = 0.3 + 1.0 = 1.3 • 100 = 102 à log(100) = 2 • 1000000 = 106 à log(1000000) = 6 How to use a logarithm • As a very simple example, what is log(x) = 4? • Log(x) = y à x = 10y è log(x) = 4 à x = 104 = 10000 • For example, the mass of the sun in log is log(Msun)=30.3 in units of kg • What is this this is real numbers: • Remember: log(x) = y à x = 10y • so in our example: • Log(M) = 30.3 à M = 1030.3 = 100.3 x 1030 =2x1030 kg Example for a calculation • What is the mass of the sun? • We will see later that a mass can be calculated by:

• Where M is the mass of the sun, v the velocity of the Earth around the sun (v=30000 m s-1) , r the distance between the Earth and the sun (r=1.5 x 1011 m) and G is the gravitational constant G = 6.67 x 10-11 m3 s-2 kg-1 • So then the equation above becomes: • Have fun with this J M = v2r G M =300002 ×1.5×1011 6.674×10−11 m2m s2m3s2kg=2×1030kg Example for a calculation • To make things easier, do the equation • In logarithms: • log(M) = 2 x log(30000) + log(1.5 x 1011) – log (7 x 10-11) • log(M) = 2 x 4.477 + 11.2 – (0.845 – 11) = 8.95 + 11.2 + 10.155 = 30.3 • So we know that 0.3 is log(2), so the 30.3 is equal to log(2) + log(1030). • The the mass of the sun will be 2 x 1030 kg • This type of calculation also helps you to verify your result if you are in the right ballpark M =300002 ×1.5×1011 6.674×10−11 m2m s2m3s2kg=2×1030kg ...