Improving the fit to the line

# Halogen Lamp

# Are Poisson?

# Mapping wavelength with pixel position and micrometer position

# ---+ Standard Deviation vs Exposure time

# Vega

- Initially a gaussian convoluted with a circular distribution was fitted to the spectral line, with the equations of:
- Gaussian:

- Circle:

- This gave a plot of

- Then for the background term a quadratic was used of the form:

- This gave a plot of:

After this a skewed Gaussian was fitted to the spectral line:

- This has the equation:

- where

- Finally the quadratic background was fitted with the skewed gaussian:

- The parameters for each of the its is listed below:

Parameters | Conv | Conv+Quadratic back | Skew Guass | Skew Gauss + Quadratic back |

Gauss Amp | 364.6 ±(1.8e7) | 347.8 ±(4.8e6) | (3.4±160569)e05 | (6.23±1090)e4 |

Mean | 59.41±0.0003 | 33.4±0.001 | 777±1729 | -8.52±14871 |

Gauss | 1.998±0.001 | 1.785±0.002 | 75.1±3489 | 1.354±2369 |

Circle radius | 4.15±0.004 | 4.516±0.002 | ||

Circle Amplitude | 369.8±(1.8e7) | 388.2±(5.4e6) | ||

background | 624.7±0.2 | 612±0.3 | ||

1295±931461 | 0.9306±679 | |||

-5.69±2644 | -0.108±19.1 | |||

0.31±1407 | -0.005±9.4 | |||

681±0.41 | 63.2±920 | |||

-0.366±0.009 | 0.126±480 | |||

0.096±0.001 | -0.036±0.006 | |||

191158 | 138587 | 131805 | 127977 | |

70 | 70 | 70 | 70 | |

per DOF | 2730 | 1979 | 1882 | 1828 |

- it was noticed that towards the edges of the frame the distributions were becoming skewed, this was symmetric on both sides of the chip as the plots below show:

- This shows a skew on the left side of the chip:

- This shows a skew to the right of the chip:

- This shows the same peak in the center

- To look at the response of the camera a halogen lamp was observed across its spectrum, this gave a plot of:

- Wanted to check whether actually followed Poisson distribution, as the errors seemed too small
- Took 20 frames of the same spectral line. Then studied the same point for each of the 20 frames to see how it fluctuated in each frame.
- Plotted against the pixel position #350 for each of the 20 frames, and calculated the and for the 20 points

Mean | Standard Deviation |

138928.0 | 623.85 |

- For a Poisson distribution, the standard deviation is equal to the square root of the mean, . For 138928 photoelectrons this corresponds to = 372.73
- This indicates that is NOT in fact Poisson distributed, as we had been assuming before?

- First to visualise the fit to the 3d plot the residuals of the points were plotted, this is done by:

- The error bar for was calculated by propagating the errors in the pixel position and the micrometer position.
- This was done using the equation:

- where ,micrometer position and , the position on the camera.
- The first equation that was fitted was:

- This gave a residual plot of:

- This process was then repeated with a quadratic fit with the equation:

- This gave a residual histogram of:

- This was then repeated one more time with a cubic term within the fit:

- This have a histogram of residuals as:

- this gave parameters for each of the fits as:

Parameters | Linear | Squared | Cubed |

275 ± 4 | 276±9 | 314.9±23.6 | |

0.1069±0.0057 | 0.118±0.028 | 0.098±0.092 | |

55.26±0.66 | 53.76±3.57 | 26.1±15.1 | |

-(2±3)e-05 | (-1.3±2)e-04 | ||

0.11±0.34 | 5.53±3.22 | ||

0.0014±0.003 | 0.003±0.002 | ||

(1.17±1.62)e-07 | |||

-0.321±0.219 | |||

(-3±20)e-05 | |||

(-2.75±1.67)e-03 | |||

7305 | 7220 | 6772 | |

72 | 72 | 72 | |

per DOF | 101 | 100 | 94 |

* Ask about errors that go into the least squares fit, as currently set to one?

- Took frames of one spectral line of Helium at different exposure times starting at 0.1s, increasing in 0.3s increments up to 3.0s.
- This was to see how the of the line changed with exposure time
- Did fit an exponential distribution, however this is redundant without any error bars, so we need to sort out our errors before fitting it.

- The plot below shows vega at 3.5mm grating setting,
- 120s frame

-- JosephBayley - 22 Nov 2015

I | Attachment | History | Action | Size | Date | Who | Comment |
---|---|---|---|---|---|---|---|

png | PoissonTest.png | r1 | manage | 39.6 K | 23 Nov 2015 - 21:56 | AshleaKemp | |

png | SigmaError_Vs_Exposure.png | r1 | manage | 31.0 K | 25 Nov 2015 - 07:55 | AshleaKemp |

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Topic revision: r10 - 01 Dec 2015 - JosephBayley

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