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Astr 105: Aug 28, 2008
== Unit conversions ==
Changing between multiple system of units.
A measurement is a dimension (magnitude) and units.
Most different unit systems for similar measurements have the same zero-value; that's to say, 0 m = 0 ft = 0 ltyr = 0 parsecs etc. One exception to this rule is temperature.
** If we're moving between scales with zero offsets, multiply by your number system's multiplicative identity (probably exactly 1). Because a/b = 1 iff a==b in real space, we can use any equivalent units in this way to get a muliplicative identity, e.g.: 2.54 cm / 1 inch = 1.
1.61 euros/liter in $/gal:
1 euro = $1.45
1 gal = 3.78 L
therefore, 1.61 euros/liter * ($1.45/1 euro) * (1 gal/3.78 L)^-1 = $8.82/gallon
** 1 Astronomical Unit = 1 AU = 1.496 x 10^13 cm = semi-major axis of the Earth's orbit about the Sun. **
The Parsec (from the phrase parallax second):
The parallax angle = the angle between the rays from points of view from earth when it is at different places in orbit. This ray connects earth, the star, and some background object (like a distant galaxy) which isn't moving.
Suppose a right triangle such that the base is 1 AU, the other leg is _d_, and the angle between _d_ and the hypotenuse is p, the parallax angle.
Then tan(p) = 1 AU / (_d_)
We assume that for small angles,
tan(p) ≈ sin(p) ≈ p (in radians)
The parsec is defined, therefore, as:
1 parsec ≡ distance at which 1 AU subtends 1" of arc.
So, _d_ = 1 AU/( 1" in radians )
1" in radians = 1" * (1'/60") * (1 deg/60') * (2π/360 deg) ≈ 4.85 x 10^-6 (radians)
So, _d_ = 1.496 x 10^13 cm / 4.85 x 10^6
** Therefore: _d_ = 1 parsec = 1 pc = 3.086 x 10^8 cm **
1.99 * 10^33 g = Mass of the Sun... grams is an inconvenient unit!
So instead, we use:
** M_⊙ (Solar Mass Unit) = Mass of the Sun. **
Brigtness scale: uses the magnitude system, based on the human eye.
!!! The Human eye is logarithmic; that's to say, you see changes in brightness of a factor of two in even steps. !!!
Sirius has a magnitude of 0. Fainter stars are magnitudes 1 - 6ish. But they aren't equal steps in brightness, because it's based on the human eye and is therefore logarithmic.
== BACK TO COSMOLOGY ==
Four theories of universal expansion: the universe's size function is parabolic (it will collapse), it is cubic (it will expand, slow, and the expand rapidly into infinity), it is constant (it will grow continuously), or it is logarithmic (it will grow constantly but then begin to level off, approaching some value at t->inf).
Each of these theories depends on the amount of Dark Matter and Dark Energy in the universe... because these quantities are important in a universal-scale gravitational force.