Today: Our Place in the Cosmos Math Methods

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TodayOur Place in the CosmosMath Methods
Astr110A Fall 2008 Roy Gal Lecture 3
Sept. 2
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Our place in the cosmos

• The universe has structure on many different size
  scales.
• What is our

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Our place in the cosmos
Your friends, Lrrr and Ndnd, rulers of Omicron
Persei 8, want to send you a postcard via the
Cosmic Postal Service. The CPS is very strict
about complete addresses on all mail! Help them
by using all seven of the terms in the box to
complete your cosmic address put them in order,
from smallest to largest
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Our place in the cosmos
Earth The Local Group The Milky Way The Orion
Arm The Solar System The Universe The Virgo
Supercluster
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• Earth
• radius 6370 km
• distance to sun 149,598,000 km
• 1.49598x109 km 1AU
• Solar System
• distance to Neptune 30 AU

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• Orion Arm
• Milky Way Galaxy
• disk radius 3.162 x109 AU
• 50,000 light-years
• disk thickness 1,000 ly

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• Local Group
• distance to M31
• 2.5x106 ly
• Virgo Supercluster
• diameter 2 x 108 ly
• contains 2000 galaxies

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• The Universe
• 9.3 x 1010 ly across
• 13.3 billion years old

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The fun part Math Methods

• Math essentials
• -Algebra
• -Scientific Notation
• -Exponentiation
• -Logarithms
• -Significant Figures

Astronomy related -Small angles -Scaling -Uncerta
inties -Statistics
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Why I am causing (some of) you pain

• Mathematics is the language of the Universe.
  Imagine learning English and not being able to
  use any vowels
• mgn lrnng nglsh nd nt bng bl t s n vwls
• or being given a coded message without a
  key
• ?????????????????????????????
• Similarly, astronomy cannot be taught
  adequately without using the proper language.
  Some basic astronomy concepts can be explained
  best and most completely by the use of a formula.
• 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.

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Procedure

• The class will count off A and B partner with
  someone of the other letter
• Your worksheets contain examples of math
  techniques we will use.
• I will briefly review each technique with the
  class.
• Complete the question(s) for that technique
  marked with your letter (A/B)
• Share and discuss answers with your partner
• We will go over the correct answers

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Algebra

• The key to solving simple algebraic equations
  containing a single unknown (e.g. x 6 10) is
  to realize that the equation is an equality. As
  long as you do the same mathematical operation
  (e.g. add or subtract a constant, multiply or
  divide by a constant, squaring, taking a root) to
  both sides of the equation, the equality is still
  an equality.
• Distributive Law
• 3(x 2) 3x (3)(2) 3x 6
• Associative Law
• 4x - 7x x(4 - 7) -3x

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Algebra Examples

• x610 Solve for x
• Solution subtract 6 from both sides
• x4
• 2x-6 -3 Solve for x
• Solution add 6 to both sides, divide by 2
• 2x 3 x 3/2
• 49 (3x8)2 / x
• Trickier requires quadratic formula.
• Rework into form ax2 bx c 0
• Solutions

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• 1.1. Algebra
• A. Solve for y    x 4y2    
• B. Solve for m (mass)  E  mc2      
• 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)2

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Scientific Notation

• Used for very large and very small numbers
• Two terms, digit and exponent
• 1,500,000,000,000 1.5 trillion 1.5 x 1012
• the exponent of 10 is the number of places the
  decimal point must be shifted to give the number
  in long form.
• Positive exponent decimal place is shifted to
  right
• Negative exponent Decimal place shifted to left
  1.5x10-4 0.00015

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• 1.2. Scientific Notation
• Write the following in scientific notation
• A. 3,042 3.042 x 103
• B. 231.4 2.314 x 102
• A. 0.00012 1.2 x 10-4
• Convert the following numbers from scientific
  notation
• B. 4.2 x 109 4,200,000,000
• A. 4 x 10-6 0.000004

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Math with exponents

• Adding/subtracting
• All numbers are converted to the same power of
  10, and the digit terms are added or subtracted.
• Example (4.215 x 10-2) (3.2 x 10 -4)
• (4.215 x 10-2) (0.032 x 10-2)
  4.247 x 10-2
• Multiplication
• The digit terms are multiplied in the normal
  way and the exponents are added. The end result
  is formatted so that there is only one nonzero
  digit to the left of the decimal.
• Example (3.4 x 106)(4.2 x 103) (3.4)(4.2) x 10
  (63) 14.28 x 109 1.4 x 1010 (to 2
  significant figures - will discuss later)

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Math with exponents

• Division
• The digit terms are divided in the normal way
  and the exponents are subtracted. The quotient is
  changed (if necessary) so that there is only one
  nonzero digit to the left of the decimal.
• Example (6.4 x 106)/(8.9 x 102) (6.4)/(8.9) x
  10(6-2) 0.719 x 104 7.2 x 103 (to 2
  significant figures)
• Powers of Exponentials
• The digit term is raised to the indicated
  power and the exponent is multiplied by the
  number that indicates the power.
• Example (2.4 x 104)3 (2.4)3 x 10 (4x3)
  13.824 x 1012 1.4 x 1012

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Logarithms

• Inverse of exponentiation
• The log of a number tells us the equivalent
  number that we raise 10 to. For example
• -The log of 1000 is 3 because 103 is 1000.
• -The log of 6000 is approximately 3.778
  because 103.778 is (about) 6000.
• We know that the log of 6000 would be between 3
  and 4 because 103 is 1000 and 104 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 10x button to
  reverse the process.

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• 1.4. Logarithms
• Solve for M in the equation M m - 5 log (d) 5
  when
• A
• B
• A

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Significant Figures

• All non-zero digits are considered significant.
  Example 123.45 has five significant figures 1,
  2, 3, 4 and 5.
• Zeros appearing anywhere between two non-zero
  digits are significant. Example 101.12 has five
  significant figures 1, 0, 1, 1 and 2.
• Leading zeros are not significant. For example,
  0.00012 has two significant figures 1 and 2.
• Trailing zeros in a number containing a decimal
  point are significant. For example, 12.2300 has
  six significant figures 1, 2, 2, 3, 0 and 0. The
  number 0.00122300 still has only six significant
  figures (the zeros before the 1 are not
  significant).
• The significance of trailing zeros in a number
  not containing a decimal point is unclear. Is
  1300 accurate to the nearest unit (and just
  happens coincidentally to be an exact multiple of
  a hundred) or if it is only shown to the nearest
  hundred due to rounding or uncertainty. Clarify
  as 1300.

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• 1.5. 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.
• 1.5.1. How many significant digits are in the
  following numbers?
• B 1.5 _____
• A 3.5689 _____
• B 4000 _____
• A 3.68 _____
• 1.5.2. Multiply 1.5, 3.5689, 4000, and 3.68.
  Round your answer to the correct number of
  significant digits.
• B 1.5 x 3.5689 x 4000 x 3.68 ________

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Small angles

• Many angles in astronomy are very small (degree)
• Examples parallax, apparent sizes of planets and
  distant galaxies
• Can avoid trigonometry
• S ?
• D
• sin(?) S/D
• For small angles, sin(?) tan(?) ?

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Small angles

• If we know the distance to an object, and measure
  its angular size, we can calculate its actual
  size
• Conversely, if we measure the angular size, know
  the actual size, we can get distance