# Astronomy 217

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