Astronomy 217

 
 

Prof. Andrew W. Steiner

 
 
Aug. 20, 2021
 
 

TA Bryce Fennig

Outline

  • Syllabus
  • Office Hour
  • Scientific Notation
  • Units
  • Trigonometry
  • Applying to stars
  • Taylor series

Scientific Notation

  • \( \mathrm{mantissa} \times 10^{\mathrm{exponent}} \), e.g. \( -1.37 \times 10^{-10} \)
  • The exponent tells you how many places you need to move the decimal point. $$ 3.4 \times 10^{4} = 34,000 $$ $$ 1.2 \times 10^{-3} = 0.0012 $$
  • Moving decimal points: $$ -1.37 \times 10^{-10} = -0.137 \times 10^{-9} = -13.7 \times 10^{-11} $$
  • It is typical to always move the decimal point to ensure there is only one digit to the left

Arithmetic in Scientific Notation

  • To add, move the decimal points to make the exponents equal, and then add mantissas $$ 1.34 \times 10^{2} + 6.5 \times 10^{-1} $$ $$ = 1340 \times 10^{-1} + 6.5 \times 10^{-1} $$ $$ = 1346.5 \times 10^{-1} = 134.65 $$
  • To multiply, multiply mantissas, add exponents, and then move the decimal point if necessary $$ 2.5 \times 10^{2} + 6.5 \times 10^{-1} $$ $$ = 16.25 \times 10^{(2\,+\,-1)} $$ $$ = 16.25 \times 10^{1} $$ $$ = 1.625 \times 10^{2} = 162.5 $$

Units

  • very important
  • Everyday units: pounds, feet, yards, etc.
  • Système international: meters, kilograms, seconds, Kelvin
  • Astronomy: centimeters, grams, seconds, Kelvin
    year, light year, astronomical unit, parsec, Bethe
  • Nuclear physics units: Fermis, MeV, fm/c, MeV
  • Units I use: femtometers, inverse femtometers, femtometers, inverse femtometers
  • "Natural units": 1, 1, 1, 1

Arithmetic with Units

  • You can only add or subtract quantities with the same units $$ 2~\mathrm{ft}+3~\mathrm{lb} = \mathrm{disaster} $$ $$ 2~\mathrm{kg}+3~\mathrm{kg} = 5~\mathrm{kg} $$
  • When multiplying or dividing quantities with units, multiply or divide both the numbers and the units $$ 2~\mathrm{kg} \times 3~\mathrm{m} = 6~\mathrm{kg~m} $$ $$ 2~\mathrm{kg} \div 3~\mathrm{m} = 6.67 \times 10^{-1}~\mathrm{kg/m} $$
  • You cannot use quantities with units in exponential and logarithmic functions $$ \exp (2~\mathrm{meters}) = \mathrm{disaster} $$
  • You cannot use quantities with non-angular units in trigonometric functions $$ \sin (2~\mathrm{meters}) = \mathrm{disaster} $$

Trigonometry

$$ \begin{eqnarray} x^2 + y^2 = z^2 \\ \sin^2 \theta + \cos^2 \theta = 1 \end{eqnarray} $$  
$$ \begin{eqnarray} x = r \cos \alpha \\ y = r \sin \alpha \\ y/x = \tan \alpha \\ \end{eqnarray} $$  
$$ \begin{eqnarray} \sin \alpha = \sin (\alpha + 360 \deg) \\ \sin \alpha = \sin (\alpha + 2 \pi) \\ \cos \alpha = \cos (\alpha + 360 \deg) \\ \cos \alpha = \cos (\alpha + 2 \pi) \\ \tan \alpha = \tan (\alpha + 360 \deg) \\ \tan \alpha = \tan (\alpha + 2 \pi) \\ \end{eqnarray} $$  
$$ \begin{eqnarray} \cos \alpha = \sin (\alpha + \pi/2) \\ \end{eqnarray} $$

Units for Small Angles

  • 360 degrees is \( 2 \pi \) radians
  •  
  • 1 degree is 60 arcminutes
  • 1 arcminute is 60 arcseconds, aka "seconds of arc"
  • \( 360 \times 60 \times 60 = 1.296 \times 10^6 \) arcseconds in a full circle
  • \( 2.0626 \times 10^5 \) arcseconds in a radian
  •  
  • 360 degrees is 24 hours
  • 1 hour is 60 minutes
  • 1 minute is 60 seconds
  • \( 24 \times 60 \times 60 = 8.64 \times 10^4 \) seconds in a full circle
  • \( 1.375 \times 10^4 \) seconds in a radian

Taylor Series

  • The following holds for \( x \) near \( x_0 \): $$ f(x) = f(x_0) + (x-x_0) f^{\prime}(x_0) + \frac{(x-x_0)^2}{2!} f^{\prime\prime}(x_0) + \frac{(x-x_0)^3}{3!} f^{\prime\prime\prime}(x_0) $$ where, e.g. \( f^{\prime \prime}(x_0) \) means take the second derivative of the function \( f(x) \) and evaluate it at \( x= x_0 \).
  • E.g. $$ \exp x \approx 1 + x + \frac{x^2}{2} + \ldots $$ $$ \sin x \approx x - \frac{x^3}{6} + \ldots $$ $$ \cos x \approx 1 - \frac{x^2}{2} + \ldots $$ (but the last two work only for radians!)

Stellar trigonometry

  • Angular separation between GQ Lupi and its largest exoplanet is 0.732 arc seconds
  • How many degrees is that?
  • How many radians?
  • If the distance to GQ Lupi is 495 light years, what is the distance between GQ Lupi and its largest exoplanet in light years?

Derivatives

  • The derivative of f is $$ \lim_{\Delta x \rightarrow 0} \frac{\Delta f}{\Delta x} \equiv \frac{df}{dx} $$
  • The derivative is the slope of the line tangent to the curve

Mechanics and Calculus

  • Position as a function of time \( x(t) \)
  • Velocity as a function of time $$ v(t) = x^{\prime}(t) = \frac{dx}{dt} $$
  • Acceleration as a function of time $$ a(t) = x^{\prime \prime}(t) = v^{\prime}(t) = \frac{d^2x}{dt^2} = \frac{dv}{dt} $$
  • Momentum (classical), \( p = m v \)
  • Newtonian mechanics: $$ F = m a = \frac{dp}{dt} $$ (assuming the mass is constant)

Constellations

  • Use in antiquity
  • Now 88 official regions in the sky
  • Borders follow lines of right ascension and declination
  • Shapes drawn out by stars are "asterisms"
  • Within constellations, stars are labeled with greek letters by brightness
  • Objects measured in radio or X-ray use a different naming scheme

Supplement: Steiner's Laws of Astronomy

  • Nature makes almost everything we can think of, and some things we haven't thought of yet
  • Space is big; distances are hard
  • Space is dirty
  • Everything rotates: angular momentum conservation, energy conservation, and gravity force astrophysical objects into a common theme with bulges, disks, and jets
  • Scientists are bad at naming things
  • Almost everything is nearly a blackbody. Nothing is exactly a blackbody