ASTRONOMICAL DATA ANALYSIS

ASTRONOMICAL DATA ANALYSIS

Andrew Collier [email protected]

Text: Press et al, Numerical Recipes

Astronomical Data• (Almost) all information available to us about

the Universe arrives as photons.• Photon properties:

– Positionx– 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()

– 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.

QuickTime™ and aVideo decompressor

are needed to see this picture.

Doppler tomography f(, t)

Starspotsignatures inphotospheric lines

-v sin i +v sin i

Starspots

Prominencesignatures inH 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.

QuickTime™ and aVideo decompressor

are needed to see this picture.

Latit

ude

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.

0

20

40

60

80

100

120

140

-20 -10 0 10 20

• 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”?

Random variables

-4

-2

0

2

4

0 10 20 30

-4

-2

0

2

4

0 10 20 30

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)a

b

∫ x ≡P(< x ≤b)

f (x)−∞

∞

∫ x =1

f(x)

xa b

Cumulative probability distributions

f(x)

xaF(a)≡ f(x)

−∞

∫ x ≡P(x ≤)

F(∞)=1F(−∞)=0

F(x)

xa0

1

• Integrate PDF to get probability that x ≤ a:

Discrete probability distributions• e.g.

– Exam marks– Photons per pixel

f (x)≡ ii∑ (x−xi)

F(b)≡ ii∑ forxi ≤b

Histogram

0

0.5

1

1.5

2

2.5

3

3.5

4

-0.504.509.5014.5019.5024.5029.5034.5039.5044.5049.5054.5059.5064.5069.5074.5079.5084.5089.5094.5099.50Bin

Frequency

.00%10.00%20.00%30.00%40.00%50.00%60.00%70.00%80.00%90.00%100.00%

FrequencyCumulative %

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

f (x)= 1b−

for< x < b

f(x)=0otherwise.

x

U(a,b)

a b