How do we see?

• We see an object when light from that object reaches our eyes
• Either because the object itself emits light
• Or because that object reflected light

Some Light Processing

• We get two pieces of information:
  • The direction the light enters the eyes gives us positional information
  • The color of the light gives us information about the composition of the light

Tricks of Light

• A single image gives no distance information
• Light can bend, which can confuse your brain

But what IS it?

• Light is, first and foremost, a wave
  • Similar to an ocean wave, in that it travels in a direction
  • Affects both electric charges and magnets “floating” atop itself
    • Accordingly referred to as an electromagnetic wave

Properties of Waves

• All light waves move at the speed of light, \(c\): \[ c = 3\times 10^8 \text{ m/s}\]
• Amplitude corresponds to brightness
• Wavelength corresponds to color or portion of spectrum

The Rainbow

You’re (Mostly) Blind!

Everything the light touches…

• The general term electromagnetic radiation describes all the wavelengths of electromagnetic waves, not just the visible ones we generally refer to as “light”
• Everything that emits or reflects radiation we can observe
• The visible bits are just a tiny fraction of the huge spectrum of possibilities
• Gives rise to different forms of astronomy:
  • Optical
  • Radio
  • Microwave
  • High Energy (Gamma/X Ray)

Recipe: How to make some light

• How does one produce electromagnetic radiation?
  • Microscopically: by accelerating electric charges
  • Macroscopically: by making something hot
    • By “hot” we just really mean “not 100% cold”…
    • Heat excites the particles, bouncing them around and thus accelerating charges
• What wavelengths are emitted depends on the object’s temperature
  • Hot objects produce more radiation, so greater amplitudes overall
  • Hot objects produce more radiation at shorter wavelengths

A Shining Example

Planck’s Law

• The brightness at a given wavelength emitted from a body in thermal equilibrium at some temperature is governed by Planck’s Law: \[ B(\lambda, T) = \frac{2hc^2}{\lambda^5} \frac{1}{\exp\left(\frac{hc}{\lambda k_B T}\right) - 1} \] where \[ \begin{aligned} h &= 6.6261 \times 10^{-34} \text{ J}\cdot\text{s}\\ c &= 3\times10^8 \text{ m/s} \\ k_B &= 1.381 \times 10^{-23} \text{ J/K}\\ \lambda &= \text{wavelength in meters} \\ T &= \text{temperature in kelvin} \end{aligned} \]

Planck’s Law Visualized

Nothing is Perfect

Wien’s Law

• Wien’s Law relates the wavelength at the peak of the spectral black body curve to a temperature
• Can be useful if you only care about the temperature, and can observe the peak of the curve
• Has a very simple expression: \[ \lambda_{peak} = \frac{2.8977\times10^{-3}\text{ m}\cdot\text{K}}{T} \] in standard units, where \(T\) is measured in kelvin

Fitting Planck

• A spectra is an observable quantity
  • Even if the star is far away so that it’s brightness is lower, that just scales down the entire curve
• Spectra can thus be used to determine the approximate temperature of stars though either:
  • Use of Wien’s Law
  • Fitting Plancks law directly
• Use of Wien’s Law is usually simpler, but there can be times when you need to fit the entire blackbody curve
  • Requires a nonlinear fit, but can still be done using common least squares algorithms (at least for the precision we need here)

Least Squares Nonlinear Fitting (Python)

• In Python, you want curve_fit from scipy.optimize for this probably (docs here)
• Need to define the function you want to fit, where the first parameter is the independent variable, and subsequent parameters are any desired fit parameters
  • For Planck’s law, that means wavelength is the first parameter, and amplitude and temperature the second
• When using curve_fit need to provide:
  • The function name you want to fit
  • The xdata
  • The ydata
  • An initial guess for any fit parameters (else starts at 1)

Least Squares Nonlinear Fitting (R)

• In R, you’ll likely want to use the nls function

• Give it a formula, using the column names where appropriate:

  brightness ~ A / wavelength^5 ...

• Specify what dataframe you are pulling the column names from:

  data=df

• Need to provide a list of starting values for the parameters

  start = list(e=100, T=1000)

Fitting Demos

• For a demonstration, we are going to use the data here
  • Noisy black body data with some general reduction in brightness
  • Want to fit both the temperature and the emissivity
• We can compare our fits to what we’d have gotten from Wein’s law

New Unit, new Groups

• While you may still be finishing up HW2 with your past partners, I wanted to make you aware of the new partnerships, so that you can make arrangements for this week