Download The effect of seeing on Stellar Profiles As light passes through the

The effect of seeing on Stellar Profiles
As light passes through the Earth’s atmosphere, the cells refract light by different amounts due
to the varying temperatures and pressure within each cell. The effect is to blur a point source
such as a star in to a smoothly varying disc. If the image is oversampled, that is, if the resolution
of the camera is greater than the typical seeing disc, then we can model the stellar profile as a
2d Gaussian function. Assuming that there is no loss of photons (there is, but we can ignore it
for the purpose of this doc), then if we count all photons in the stellar disc then we must
recover the amount before the photons reached the atmosphere. Clearly then the bigger the
seeing disc, the worse the seeing, and consequently the lower the photon arrival rate
(photons/second) at each pixel in the image.
The question we ask ourselves is: What is the probability a photon will end up in a given pixel,
based on the seeing conditions and the resolution of the camera?
As mentioned previously, we can model the distorted star image as a 2D Gaussian profile. That
is to say, the probability a photon will arrive at a given location is given by a 2D Gaussian PDF
(probability density function). In 1D this is:
where μx is the position of the maximum of this function (the mean value), and σx is the
standard deviation of the stellar profile. It is the value of the standard deviation that
determines how blurred the image is. In 2D the Gaussian PDF is:
We can readily simplify this by using the following approximations. Firstly the centre of the star
occurs at the same location in both the x and y directions, so μ x= μy. Also the star image can
very safely assumed to have circular isophotes (places of equal brightness), so we can safely say
that σx= σy. That is the star is equally blurred in all directions, and appears circular. Further
more, we can define the centre of the star image as the (0,0) point, so that μx= μy=0. The 2D
Gaussian PDF becomes:
Now the seeing is usually characterised by the FWHM (Full Width Half Max) of the star image.
The FWHM, F, is related to the standard deviation by:
Solving for σ and squaring gives:
Substituting in for σ2 in the 2D Gaussian PDF gives:
This PDF is sampled by a pixellated detector, so to find the probability of a photon landing
within a given pixel we must integrate over the size of the pixels. We assume that the centre of
the star image is located within the centre of the pixel. This is not quite true (and is a big
problem for astrometry) but it is very close.
Each pixel sees a patch of sky, R, the resolution, given by
where p is the pixel size (m) and f is the focal length (m). R has units of arcseconds/pixel.
To work out the probability of a photon landing between x1and x2 and between y1 and y2 is:
Please excuse the x2,y2 etc notation, but excel won’t allow me to enter another set of limits for
the y integration.
The above equation is the most general expression, and is the probability of finding a photon
between the points x1,x2 and y1,y2.
The centre pixel will gather most photons, so what is the probability that a photon will land
there, given a value of F and R?? We previously said that the centre of the star profile was the
centre of the pixel, so the limits of the x and y integration are:
x1=-R/2; x2=R/2; y1=-R/2; y2=-R/2. R/2 is just the distance from the centre of the pixel to the
edge in all 4 directions. We also previously invoked symmetry in x and y. So the probability in
the y direction is the same as the x direction. So we only have to do one integration and square
the result. Thus we need to calculate:
This integral is messy, but the result well known. It is the error function, erf(x). This integral can
be done using the internet, and the result, after simplifying is:
One further simplification exists. The error function is odd, that is erf(-x)=-erf(x). So we can
write the final probability of a photon reaching the centre pixel as:
We see that the probability is dependent on the ratio R/F. A high value for this will put more
photons in the centre pixel. This condition depends on low resolution and good seeing. If one
oversamples then the probability falls.
Lets estimate the probability for a typical set up, using value of R=1”/pixel resolution in 3”
FWHM seeing. The probability of a photon striking the centre pixel is then:
So for a fairly typical set up, only 5% of the incident photons end up in the centre pixel.
We can also note however that because the stellar profile is circularly symmetric a natural
choice of coordinates is not (x,y) but polar coordinates (r,θ). Going back to the probability
integral we can use the following relations to convert to polar coordinates:
The probability integral becomes:
Because r and θ are not dependent on each other, we can split up the integral:
where θ is the angle from the origin we wish to integrate over. Normally one is concerned with
the entire circle of radius r, and so the limits of the θ integration is 0-2π. The integral of dθ over
this range is simply 2π. Thus:
Thankfully this integral can be solved analytically. The solution is:
This is the probability of finding a photon between r1 and r2, where r is the distance from the
centre of the star in arcseconds. We can express r in the form of the radius in pixels multiplied
by the resolution of each pixel. That is:
where n is the number of pixels and R is the resolution of each pixel as previously defined. Thus
the probability is:
The above expression is useful if you want to know how many photons will be collected within
a certain radius from the centre of the star. Such a quantity appears in some programmes as
‘encircled energy’ which can be used for focussing. Normally defined as the radius within which
a certain fraction of the total light is contained. This should be as small as possible. The
probability in (x,y) is useful as even though the star exhibits circular symmetry it is then
sampled by a detector that works in the (x,y) plane. The probability of getting a photon in a
pixel is given by the expression in (x,y).
Shown on the next few pages are some graphs showing how the probability of a photon hitting
the centre pixel varies with seeing and how the total probability of detecting a photon within a
given radius varies with seeing. Also plotted are how the size of the star image that contains
80% of the collected photons varies with seeing and how much of the total light is collected
within a 2 pixel radius from the centre. An arrow marks the various values at the ‘Nyquist’ limit,
which roughly states that you should image at a resolution of half the FWHM.
The above analysis has assumed that the stellar profiles (or shapes) are approximately Gaussian
in nature. In reality this is seen to be true if the resolution is significantly more than the seeing.
However once the seeing and the resolution are about equal, the star image is affected by
diffraction by the telescope aperture, and this creates the famous airy disc. An exact description
of the probability of a photon landing in a certain pixel therefore must include the effects of
diffraction. This adds significant complexity to the problem, but in principle, it is perfectly
calculable. I intend to try to include the diffraction effects to enhance the accuracy of the
probability measurements. Despite neglecting diffraction, the profile of stars in images that are
reasonably well sampled still appear to be very close to Gaussian across most of the star image,
and so this analysis does hold a significant degree of accuracy, I believe.
Paul Kent 09/02/11