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2/25

X-ray observations of the Sun by RHESSI

background

Signal


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T ~ 3K

The CMBR is very uniform


4/25

CMBR: all-sky temperature map from WMAP satellite (Feb 2003).

Temperature fluctuations of about ,

~380,000 years after the Big Bang

K10 5


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6/25

Poisson statistics

For much of the E-M spectrum astronomical observations involve countingphotons. However, the number of photons arriving at our telescope from a

given source will fluctuate.

We can treat the arrival rate of photons statistically, which (roughlyspeaking) means that we can calculate the average number of photons

which we expect to arrive in a given time interval.

We make certain assumptions (axioms):

If our observed photons satisfy these axioms, then they are said to follow

a Poisson distribution

1. Photons arrive independently in time

2. Average photon arrival rate is a constant


7/25

Suppose the (assumed constant) mean photon

arrival rate is photons per second.

If we observe for an exposure time seconds,

then we expect to receive photons in that time.

We refer to this as the expectation value of the

number of photons, written as

R

R

RNNE ==)( (4.1)


8/25

Suppose the (assumed constant) mean photon

arrival rate is photons per second.

If we observe for an exposure time seconds,

then we expect to receive photons in that time.

We refer to this as the expectation value of the

number of photons, written as

If we made a series of observations, each of time seconds, wewouldnt expect to receive photons every time, but the

average number of counts should equal

(in fact this is how we can estimate the value of the rate )

R

R

RNNE ==)( (4.1)

RN =

N

R


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Given the two Poisson axioms, we can show that the

probability of receiving photons in time is

given byN

( )!

)(N

eRNp

RN

= (4.2)


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( )!

)(N

eRNpRN

=

N

)(Np

Poisson statistics

5.0=R

0.1=

R

0.5=R

As increases, the shape of the Poisson distribution becomes

more symmetrical (it tends to a normal, or Gaussian, distribution)

R


11/25

Poisson statistics

For the purposes of A2, all we need to work with is the mean or

expectation value of , and its variance.

We already defined the expectation value as

We can compute the mean value of using eq. (4.2),

and this confirmseq. (4.1).

N

RNE =)(

N


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We can also define the varianceof , which is a measure of the

spread in the distribution:

Poisson statistics

N

RNE =)(

N

N

( ) [ ]22 )(var NENEN == (4.3)


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We can also define the varianceof , which is a measure of the

spread in the distribution:

We can think of the variance as the mean squared error in

Poisson statistics

N

RNE =)(

N

N

( ) [ ]22 )(var NENEN == (4.3)

N


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For a Poisson distribution, the variance of can be shown to be

and the standard deviation of is

In practice we usually only observe for one period of (say)

seconds, during which time we receive (say) a count ofphotons.

We estimate the arrival rate as

Poisson statistics

N

RN =)(var (4.4)

N R=

obsN

obs N

R=

(4.5)

(4.6)


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obsN

We take as our best estimate for with error

i.e we quote our experimental result for the number count ofphotons in time interval as

Poisson statistics

obsN N obsN

obsN

(4.7)


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Usually there are several sources of noise in our observation,

each with a different variance.

Probability theory tells us that, if the sources of noise are allindependent, then we work out the total noise by adding

together the variances:

Adding Noise

2

other

2

Poisson

2

total += (4.8)


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Sources of Poisson noise:

1) fluctuations in photon count from the sky

2) dark current: thermal fluctuations in a CCD

Adding Noise

2

other

2

Poisson

2

total += (4.8)


18/25





Sources of non-Poisson noise:

1) Readout noise. e.g. a CCD can gain/lose electrons

during readout. Usually constant

Adding Noise

2

other

2

Poisson

2

total += (4.8)

=Readout


19/25

Correcting for combined quantum efficiency of telescope and

detector

Noise and Telescope / Detector Design

0tot

h

ASN

(4.11)

Fraction of incident photons that

produce a response in the detector


20/25

Suppose we observe a point source, of flux density ,

through a telescope with collecting area for time , in

bandwidth centred on .

Total energy collected by detector (ignoring any absorption)

No. of photons collected is


S

0

= ASEtot (4.9)

0

tot

h

ASN

(4.10)


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detector

Thus and


0tot

h

ASN

(4.11)



totPoisson N= ( )2/1

ASNR

(4.12)


22/25


detector

Thus and

Same result true for radio observations , even though we dont

count photons. Here, noise from source and detector electronics


0tot

h

ASN

(4.11)



totPoisson N= ( )2/1

ASNR

(4.12)


23/25


24/25

Eq. (4.12) suggests that we can increase the signal-to-noise-ratio

by increasing the bandwidth of our observation.

Line Sources

This is not necessarily true if weare observing a source which emits

only over a narrow frequency range

e.g. a spectral line.

Increasing beyond the line width

will increase the amount of noise

(from the background continuum)without further increasing the

amount of signal (from the line).

See A2 Theoretical Astrophysics for more on linewidths

frequency

int

ensity

continuum

emission line

line width


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The 2-D image of a faint galaxy observed by a CCD covers 50 pixels. For an

exposure of 5 seconds a total of 104 photo-electrons are recorded by the CCD

from these pixels. An adjacent section of the CCD, covering 2500 pixels, records

the background sky count. During the same exposure time a total of 105 photo-

electrons are recorded from the adjacent section. Show that, after subtractingthe background sky count, the signal-to-noise ratio for the detection of the

galaxy is estimated to be 73.

Calculate the length of exposure required to increase the signal-to-noise ratio to

100.

Example

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