Edmund Halley 1718 discovered that some

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Edmund Halley (1718) discovered that some stars
exhibit proper motion
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Fig.16.02b
The fastest-moving star Barnards Star 4 in
only 22 earth years!!
Willam Herschel (1783) discovered that the sun is
moving in the direction of Hercules -- called the
apex direction
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The Power and the Logarithm

• rules for exponents or powers
• xa x is the base and a is the power
• crudely, multiply x by x, repeat a times
• works for any number x or a, not just integer
  example if a ½, x1/2 ?x
• xaxb x(ab) xa/xb x(a-b) (xa)b
  xab

How Do We Undo the Power Operation?
Example suppose 10x 1000 solve for x
The inverse function is the (base-10)
logarithm So we say x log10 (1000) 3 in this
example
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Rules for Logarithms

• log (ab) log a log b Proof
  exponentiate both sides
• log (a/b) log a - log b Proof
  " " "
• log (ab) b log a Proof
  " " "

The power of the Logarithm

• Factors of 10 multiplication by 10 become
  addition

1 2 3 4 5 6
log x
101 102 103 104 105 106
x
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Stellar Apparent Magnitude is a measure of
Brightness

• Hipparchus 1st magnitude brightest, 6th
  magnitude dimmest
• in terms of actual received energy, also called
  brightness, 6th magnitude put out 100 times less
  energy than 1st magnitude (Herschel), by using
  variable aperture (objective diameter) telescope
  clever guy!!
• The magnitude scale is actually a logarithmic
  scale!! Cool!!
• since a decrease by 5 magnitudes ? 100 times
  increase in E
• a decrease by one magnitude gives 5?100 2.512
  times increase
• for two stars of brightness b1 and b2, where b
  is a direct measure of the energy reaching the
  eye
• and suppose these two stars are of apparent
  magnitude m1 , m2

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Some Examples

• brightest star is Sirius m - 1.4
• consider star X with m 0

• so Sirius is 3.6 times brighter than X
• your eye catches 3.6 times as many Watts from S
  as from X
• Sun m -26.5
• Full Moon m -12.5
• sun puts out 2.514 3.7 x 105 times full moon
  power
• dimmest in decent Wells telescope m 12
• dimmest for Hubble m 29