Comparison of fits

  • Before we fitted an unskewed Gaussian:
\begin{equation} f_G(x,y,\mu_x,\mu_y,\sigma_x,\sigma_y,\rho, \beta_1, \beta_2) = \frac{1}{2\pi\sigma_x\sigma_y\sqrt{(1-\rho^2)}} \exp \Bigg\{-\frac{1}{2(1-\rho^2)}\Bigg[\bigg(\frac{x-\mu_x}{\sigma_x}\bigg)^2 + \bigg(\frac{y-\mu_y}{\sigma_y}\bigg)^2 -2\rho\bigg(\frac{x-\mu_x}{\sigma_x}\bigg)\bigg(\frac{y-\mu_y}{\sigma_y}\bigg)\Bigg]\Bigg\} \end{equation} \begin{equation} f_{bg}(N,M) = \frac{1}{N \times M} \end{equation} \begin{equation} f(\alpha) = \alpha f_G + (1 - \alpha) f_{bg} \end{equation}
  • Now we fitted a skewed Gaussian:
\begin{equation} f_G(x,y,\mu_x,\mu_y,\sigma_x,\sigma_y,\rho, \beta_1, \beta_2) = \frac{1}{2\pi\sigma_x\sigma_y\sqrt{(1-\rho^2)}} \exp \Bigg\{-\frac{1}{2(1-\rho^2)}\Bigg[\bigg(\frac{x-\mu_x}{\sigma_x}\bigg)^2 + \bigg(\frac{y-\mu_y}{\sigma_y}\bigg)^2 -2\rho\bigg(\frac{x-\mu_x}{\sigma_x}\bigg)\bigg(\frac{y-\mu_y}{\sigma_y}\bigg)\Bigg]\Bigg\} \times \Bigg[1 + \rm{erf} \frac{\beta_1 x + \beta_2 y}{\sqrt{2}}\Bigg] \end{equation} \begin{equation} f_{bg}(N,M) = \frac{1}{N \times M} \end{equation} \begin{equation} f(\alpha) = \alpha f_G + (1 - \alpha) f_{bg} \end{equation} <br />
  • The fittings are shown below:
1124_513.png 1124_513_crop.png
180_591.png 180_591_crop.png
724_479.png 724_479_crop.png
  • Parameters:
  Star (1124,513) unskewed Star (1124,513) skewed Star (180,591) unskewed Star (180,591) skewed Star (724,479) unskewed Star (724,479) skewed
$\rho$ 1.43e-01 -1.91e-01 1.74e-01 2.34e-01 0.11 -0.04
$x_0$ 3.07e+01 2.92e+01 2.98e+01 3.03e+01 30.1 29.4
$y_0$ 3.07e+01 3.38e+01 3.06e+01 3.55e+01 31.5 36.3
$\sigma_x$ 3.86 4.28 4.18 4.30 4.29 4.37
$\sigma_y$ 4.00 5.38 4.43 8.02 4.77 6.97
$\alpha$ 1.41e-02 1.43e-02 1.42e-02 1.51e-02 0.80 0.80
$\beta_1$   8.65e-01   1.28e-01   0.40
$\beta_2$   -1.16   -2.62e+00   -1.51
$\chi^2$ 3262 3229 3538 3412 895118 571302
Degrees of freedom 3952 3952 3952 3952 3952 3952

$\chi^2$ Monte Carlo distribution

  • With 40 images generated the $\chi^2$ histogram did not look promising:
40_chi2.png
  • With a 150 images:
chi2.png
  • Therefore, the more images used the better the $\chi^$ follow the $\chi^2$ distribution.
  • 600 images:
chi2.png
  • And $\chi^2_\rm{reduced}$:
chi_red.png
  • Looks more like a $\chi^2$ distribution. However, there is a small peak in the middle, maybe more images will fix. Will try a 1000.
  • Below are the differences in found and generated parameters for 600 images:
alpha.png rho.png
x_0.png sigma_x.png
beta_1.png  
  • New errors on parameters:
  Error Now (600 images) Errors before (150 images)
$\alpha$ $\pm$ 0.0009 $\pm$0.0008
$\beta_1$ $\pm$ 0.0131 $\pm$ 0.0122
$\beta_2$ $\pm$ 0.0201 $\pm$ 0.0165
$\rho$ $\pm$ 0.0080 $\pm$ 0.0062
$\sigma_x$ $\pm$ 0.0125 $\pm$ 0.0086
$\sigma_y$ $\pm$ 0.0316 $\pm$ 0.0254
$x_0$ $\pm$ 0.0410 $\pm$ 0.0332
$y_0$ $\pm$ 0.0268 $\pm$ 0.0227
  • An example with errors:
  Error Now (600 images)
$\alpha$ 0.0014 $\pm$ 0.0009
$\beta_1$ 0.087 $\pm$ 0.013
$\beta_2$ -1.160 $\pm$ 0.020
$\rho$ -0.0191 $\pm$ 0.0080
$\sigma_x$ 4.28 $\pm$ 0.01
$\sigma_y$ 5.38 $\pm$ 0.03
$x_0$ 29.2 $\pm$ 0.04
$y_0$ 33.8$\pm$ 0.03

Observation 3: Spiral Cluster

  • Took images of the spiral cluster.
  • Below is a map of our image locations:
map.png
  • Once realigned should look something like:
combined_map.png
  • Images we took (lum):
our_images_map_copy.png
  • Here is one of the images we captured, alognside its background (luminensce filter):
original.png background.png
  • Ran through our finding algorithm, found 72 stars:
found.png
  • Some stars found are actually 2 stars.

Observation 2: Bias, dark and light frame evolution

  • The evolution before the cooling ring dissapeared:
timestape.png
  • All bias frames:
bias_1.png

bias_2.png

bias_3.png

bias_4.png

  • Decided to ignore first 20 images from the above plots to create the master bias.
  • On the left is master bias that takes all 100 bias frames, on the right is one ignoring first 20 bias frames:
master_bias_last_week.png master_bias.png
  • All dark frames (30s):
dark_all.png

  • All flat frames (no filter):
flat_all.png
  • Kept all frames for master flat:
  • Unlike last week, removed the master bias frame from each flat frame before building a master flat frame.
  • On the left is last week's flat frame, on the right it is the one with master bias removed before averaging:
master_flat_last_week.png master_flat.png

-- ElenaCukanovaite - 11 Feb 2016
Latex rendering error!! dvi file was not created.

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Topic revision: r4 - 12 Feb 2016 - ElenaCukanovaite

 
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