Advanced Topics in Astrophysics, Lecture 2

Radio Astronomy – Basics of Interferometry

Phil Diamond, University of Manchester

Resources

•ATNF Synthesis Imaging workshops 2003, 2006– http://www.atnf.csiro.au/whats_on/workshops/synthesis2006/prog.h

tml

•Texts–Radio Astronomy: Kraus–Tools of Radio Astronomy: Rohlfs & Wilson–Interferometry and Synthesis in Radio Astronomy,

Thompson, Moran & Swenson.–Very Long Baseline Interferometry and the VLBA: Napier,

Diamond & Zensus., ASP Conf series, Vol 82, 1995.

Outline

•Radio Imaging–Resolution

•2-element interferometer–Young’s Slits–Definitions–Spatial coherence function; van Cittert – Zernicke theorem

•Multi-element interferometer–Geometry–Imaging

•A tour of interferometers

Even the largest single-dish radio telescopes have very limited resolution

The resolution of the Effelsberg 100-metre at cm wavelenghts is comparable to the human eye, and much worse than a small optical telescope:

Radio Imaging

Imaging of the sky with a single-dish can be achieved by letting the source drift across the telescope beam and measuring the power received as a function of time. This provides a 1-D cut across the source intensity. Usually, the area of interest is measured at least twice, in orthogonal directions (sometimes referred to as “basket weaving”).

P

tTsys

Tsrc

This technique works best for very large and bright sources, it requires stable receivers. This is a technique that is often used at sub-mm wavelengths.

At cm wavelengths almost all imaging is conducted via the technique of interferometry.

Radio Imaging

Arc-minute resolution is often not good enough to resolve the detailed structure of many astrophysical objects e.g. distant galaxies, quasars etc. Radio sources can show emission on scales of arcminutes --> arcseconds --> milliarcseconds...

arcminutes

arcseconds

milliarcseconds

Radio Interferometry

Idea is simple (left): try and synthesize a huge radio telescope - 1-10000 km in extent by combining the signals of many small telescopes together and making use of the rotation of the Earth:

But as we learned in lecture 1, it is impractical to build radio (or indeed optical) telescopes much bigger than 100-metres.

==> The techniques of radio interferometry and aperture synthesis are used to obtain arcsecond, sub-arcsecond and even milliarcsecond resolution from interferometer arrays.

A 21cm HI single-dish image of the LMC with the Parkes telescope (Staveley-Smith et al. 2003)

Telescope alternately scanned in RA and Dec.

Angular resolution 15’

A 21cm HI Compact Array observation of the LMC (Kim et al. 2003)

1344 separate pointing centres mosaiced together.

Angular resolution 1’

D

B

where B is the maximum baseline (max distance between telescope elements in the array).

Back to basics... interference fringes and Young’s double slit experiment:

Two-element interferometer

Constructive interference “fringes” occurs when the path difference is an integer number of wavelengths. If d << L the path difference is just and the condition for maxima is:

where m is any +ve or -ve integer

Like wise the condition for minima is

B

“fringes”

In the case that y < < L, one can use the approximate formula:

Or as an angular measure the spacing between so-called “fringes” is just:

The spatial coordinates of constructive (yc) and destructive (yd) interference are then:

The spacing between successive constructive interference is then:

Simple Interferometer

• Signal reaches RH antenna τg = (D/c) sin θ before it reaches the LH antenna. τgis the geometrical delay.

• Voltages from two antennas are multiplied together and lowpass filtered or time-averaged (this combination is called a correlator)

• As earth rotates, the two antennas experience differing radial velocities with respect to source. If source were emitting monochromatic wave the signal would have a different Doppler shift in frequency at the two antennas

• Output of lowpass filter would be a sine wave whose frequency equalled the difference between the two Doppler-shifted frequencies

• This fringe frequency varies as the geometry of the interferometer changes.

Consider a fixed 2-element interferometer orientated east-west and pointing at one particular position on the sky:

+ - + - + - + -

The rotation of the Earth moves the source across the sky with the complex output of the interferometer depending on the alignment between the source structure and the fringes at any given time.

Note that the small source is unresolved by this fringe pattern. The larger source is resolved.

Simple Interferometer

Definitions

⎟⎠⎞

⎜⎝⎛= θ

λπ sinD2 cos F

Fringe frequency:

^sb•−=

^

cbτ

Delay:

^sb•−=

^2λπϕ b

Fringe Phase:

λ

2DRs >>

• Basic assumptions, required to ensure simplification of the problem:

– Source is in the ‘far field’, i.e. incoming waves are parallel

– Source is spatially incoherent, i.e. radiated signals from any two points on the source are uncorrelated

• Distant radio source in direction R radiates and produces a time variable electric field E(R,t)

• Correlation of field at points r1 and r2 (i.e. antenna locations) and at frequency ν is:

Spatial Coherence Function

)()(),( *2121 rrrr EEV =ν (1)

Spatial Coherence Function

• From this equation, and utilizing the assumptions that the source is spatially incoherent andlies in the far field, we can derive:

Where Iν(s) is the observed intensity of the radiation field; s is the unit vector in the direction of R; dΩ is the solid angle subtended by the radio source. Note that the equation depends only on the separation vector r1 – r2 not on the absolute locations. Therefore we can learn all about the correlation properties by holding one observation point fixed and moving the other about.

Vν is called the spatial coherence function. An interferometer measures the spatial coherence function. This equation also demonstrates the van-Cittert Zernicke theorem which states thatthe spatial coherence function is the Fourier Transform of the source brightness distribution

Ω= −⋅−∫ desIrrV crrsi /)(221

21)(),( νπνν

(2)

A Practical Interferometer

xcorrelator

Σ AccumulatorComputer

LOmixer

• Incoming signals are mixed down to baseband by local oscillator

phase rotator

• Phase rotation needed because delay corrector is not operating at RF frequency

τ variable delay τ

• The use of a finite bandwidth means that the path lengths on each arm should also be equal and so the geometric delay is calculated and removed.

F Fringe rotator

• The fringe frequency can be high, 100s of Hz in VLBI, compared with the reciprocal of the averaging time (seconds) so fringes can be smoothed out completely.

• The fringes are ‘stopped’ by means of fringe rotator which uses information on the interferometer’s geometry to determine the fringe frequency and remove it.

Source models & visibility functions (cont)

Extended or multi-compnent sources (left) produce more complicated functions e.g. Fomalont & Wright (1974)

A single, unresolved 1 Jy point source at the origin produces the following visibilities (data):

1 Jy

Amplitude

time

+1800

Phase

time

-1800

Multi-element interferometer

A two element interferometer produces a single response, r12.

A multi-element interferometer (say with N antennas) produces N(N-1)/2 unique responses:

Each interferometer pair presents its own sinusoid at a frequency proportional to the fringe angular spacing (which is usually different for each pair):

For N=4, 6 baselines responses are measured: r12, r13, r14, r23, r24, r34.

1 4

3 2

Geometry

To give better understanding, we now specify the geometry.

Let us imagine the measurements of Vν(b) to be taken entirely on a plane. Then a considerable simplification occurs if we arrange the coordinate system so one axis is normal to this plane.

Let (u,v,w) be the coordinate axes, with w normal to the plane. All distances are measured in wavelengths. Then, the components of the unit direction vector, s, are:

and for the solid angle

( ) ( )221,,,, mlmlnml −−==s

221 mldldmd −−=Ω

Direction Cosines

The unit direction vector sis defined by its projections

on the (u,v,w) axes. These

components are called the

Direction Cosines.

221)cos(

)cos()cos(

mln

ml

−−==

==

γ

βα

The baseline vector b is specified by its coordinates (u,v,w) (measured in wavelengths). In this special case, w = 0:

)0,v,u()w,v,u( λλλλλ ==b

u

v

w

s

α βγ

l mb

n

Imaging• With some manipulation we can rewrite the spatial coherence function

(eqn 2) in terms of u and v:

• Since above is a Fourier transform, it may be formally inverted:

• In practice, the spatial coherence function V is not known everywhere, but is sampled at particular places on the u-v plane. The sampling can be described by a sampling function S(u,v), which is zero where no data have been taken. Eqn above must therefore be rewritten as:

• This is often referred to as the dirty image

dxdyeyxIvuV vyuxi )(2),(),( +−∫∫= πνν

dudvevuSvuVyxI vyuxiD )(2),(),(),( +∫∫= πνν

dudvevuVyxI vyuxi )(2),(),( +∫∫= πνν

In the ideal case we measure the visibilities, V(u,v) over the entire uv-plane:

V(u,v)

And the FT of the visibilities produces an image of the source:

FTI(x,y)

Real life is different - sampling of the uv-plane is often incomplete - we can use image reconstruction algorithms to help produce images of the source.

V(u,v)

Imaging

Interferometer Arrays

Arrays of antennas are now used at all wavelengths from metre waves to the optical.

Image: ATNF

VLA in New Mexico, being upgraded to EVLA –being commissioned right now

E-MERLIN – being commissioned right now

LOFAR – being commissioned right now

VLBA

Very Long Baseline Interferometers

VERA

ALMA - Atacama Large Millimetre/Submillimetre Array -joint European/USA/Japanese project - currently under construction.

54 12-m antennas, 12 7-m antennas at 5000-m altitude! Freq coverage: 86-720 GHz

mm interferometer arrays:

Moveable antennas(!) - array configurations scale from 150 metres to 15 km.

Extended ALMA configuration Compact configuration

antenna transporter and pads

IRAM, Plateau de Bure, Francehttp://iram.fr/

Existing mm interferometer arrays:SMA (Hawaii) 8 x 6-m antennas, also links up with 2 nearby

sub-mm telescopes: JCMT and CSO operating at 345 GHz:

Sub-mm valley, Hawaii. Sub-mm valley, Hawaii.

JCMT SMA.

CSO

Max. baseline ~ 782m

CARMA: Combined Array for Mm Astronomy

SKA