ASTRONOMICAL DATA ANALYSIS Andrew Collier Cameron acc4@st-and.ac.uk Text: Press et al, Numerical Recipes

Astronomical Data (Almost) all information available to us about the Universe arrives as photons. Photon properties: Position x Time t Direction  Energy E = h = hc/ Polarization (linear, circular) Observational data are functions of (some subset of) these properties: f (x, t,p)

Observations I Direct imaging: f() Astrometry: f(, t) size structure Astrometry: f(, t) distance parallax motion proper motion visual binary orbits

Interferometry f(x, t) Uses information about wavefront arrival time and structure at different locations to infer angular structure of source. Picture: 6 cm radio map of “mini-spiral” in Sagittarius A.

Integral-field spectroscopy f() Uses close-packed array of fibres or lenslets to obtain spectra on a honeycomb grid of positions on the sky, to probe spatial and spectral structure simultaneously.

Eclipse mapping: f(t) Uses modulation of broad-band flux to infer locations and brightnesses of eclipsed structures.

Doppler tomography f(, t) Starspot signatures in photospheric lines -v sin i +v sin i Starspots Prominence signatures in H alpha -v sin i +v sin i Prominences Uses periodically changing Doppler shifts of fine structure in spectral lines to infer spatial location of structures in rotating systems

Zeeman-Doppler Imaging f(,t,p) Uses time-series spectroscopy of left and right circularly polarized light to map magnetic fields on surfaces of rotating stars. Latitude Longitude

Noise No two successive repetitions of the same observation ever produce the same result. e.g. spectral-line profile: Two main sources of noise: Quantum noise arises through the fact that we only detect a finite number of photons Thermal noise arises in system electronics or due to background sources.

Random variables Consider repetitions of identical measurements. Value of each data point jiggles around in some range Statistical error arises from random nature of measurement process. Systematic error (bias) can arise through the measurement technique itself, e.g. error in estimating background level. How can we describe this “jiggling”?

Probability density distributions Probability density function f(x) is used to define probability that x lies in range a<x≤b: Probabilities must add up to 1, i.e. if x can take any value between - and + then f(x) x a b

Cumulative probability distributions Integrate PDF to get probability that x ≤ a: f(x) x a F(x) 1 a x

Discrete probability distributions e.g. Exam marks Photons per pixel

Example: boxcar distribution U(a,b) Also known as a uniform distribution: U(a,b) x a b