1. ASTRONOMICAL DATA ANALYSIS
Andrew Collier Cameron
Text: Press et al, Numerical Recipes

2. 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)

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

4. 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.

5. 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.

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

7. 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

8. 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

9. 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.

10. 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”?

11. 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

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

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

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