Difference: 20160201MSciPhotometryAnalysisOb1part8 (1 vs. 6)

• Generated 40 images (100x100 pixels) of the same BRIGHT star.

• Generated dimmer star to see how errors scaled:

• For a dim star, fractional errors:

• An example with fitted params:

• Here is the progression of the cooling ring:

• The ring took 1hr50mins to dissapear.

• We had a major problem, for some reason the average pixel value for the bias frame, the dark frame exposure time 30s and 60s were all around 900.

• We tried some initial removal of bias and flat frames:
• Original:

• Bias master frame removed (substracted bias):

• Master flat removed (divided by master flat):

Parameter errors generated with Monte Carlo

• Generated 40 images (100x100 pixels) of the same star.
• As before generated a star according to:

• Therefore, each pixel had a distinct pixel value.
• Added a background of 160 pixel values.
• Then for each pixel, its pixel value was taken as the mean value of the Poisson distribution and a new pixel value was then generated according to that distribution for the given pixel.
• The 40 images were fitted using our maximum likelihood fitting algorithm.
• For each parameter fitted the following was calculated:

• Bellow are the histograms for the above calculation:

• Decided to take the error on the parameter as the difference between the biggest and smalles fractional errors.
• Here is the fractional errors:

• An example with fitted parameters:

Skewed Gaussian

• Since our stars looked skewed, decided to fit skewed Gaussians.
• The new total function fitted, :

\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}

%ENDLATEX%

• Below are examples of fitted stars from one of our images:

• Well fitted stars seemed to have at around 1.
• Therefore, we could set our threshold for the boundary between 1 star and 2 stars as .
• It appears that the most bright stars have been fitted terribly and their are even bigger than 2. It seems that although the tails fit nicely, the fit cannot reach the required amplitude.
• Below are histograms for the same stars in the same order of

• This should have produced a Gaussian with mean 0 and standard deviation of 1.

• As one would expect, the bigger the , the further away the histogram is from a normalised Gaussian.

Observation 2

• Took a series of bias, dark and flat frames to check for variabilities and to figure out how long it took the cooling ring to disappear.
• Took a selection of master frames for later analysis porpoises.

Checking fit validity

• p values give a probability, so should be normalised to 1.
• Found p value for each pixel, and summed over all pixels
• Found summed values as: 0.999999876234048 1

• Found previously around 80% signal and 20% background,
• Performed back of the envelope calculations to validate this, found s = 80%, b = 20% as was fitted.