This document discusses sensor modeling and photometry techniques applied to astrophotography. It presents a case study imaging the Triangulum Galaxy using a modified DSLR camera and telescope. Key steps included estimating signals and noises from light, dark, and offset frames to characterize the sensor and extract measurements of sky background, dark current, and target object luminance. Expectations and variances of photoelectrons generated were related to measured analog-to-digital units. The techniques allowed optimizing exposure times based on read noise, guiding error, and dynamic range criteria.

1. Sensor modeling and Photometry:
an application to Astrophotography
Laurent Devineau

2. Introduction
▪ The objective is to present with a case study, key notions of photometry and sensor modeling
applied to astrophotography.
▪ After an introduction to different signal and noise categories captured by a camera, we deal with
time exposure optimization of an astrophotography session .
▪ We detail methodologies based on Read Noise dominance, on guiding error and on loss of dynamics
management. This allows to get optimal time exposure regarding these different criteria.
▪ We also present signal to noise ratio (SNR) calculation techniques of a stacked image and we
formalize a methodology to determine global exposure time to reach a target SNR.
▪ Some theoretical developments are also proposed to justify the Poissonian nature of the photonic
signal and to prove the expectation / variance equality of the photoelectrons generated by the
sensor.
2

3. Case study description
▪ The techniques detailed in this document are illustrated with a
case study.
▪ This case study consisted in imaging the Triangulum Galaxy (M33)
with the following framework:
▪ Digital Single Lens Reflex (DSLR) Canon 500 D modified with a sensor
CMOS and a pixel size of 4.68 µm ;
▪ Newtonian telescope(*) of 1000 mm focal length and a primary mirror of
200 mm (𝐹/𝐷 = 5 aperture).
▪ Below the details of image series:
▪ 80 unitary light frames of 240 seconds at ISO 400, for a global exposure
time of 5h20’
▪ A series of DOFs (Darks Offsets Flats) consisting of :
• 45 dark images of 240 seconds at ISO 400
• 31 flat images of 1/3 second with a ISO 100 gain (in order to
concentrate the histogram on 2/3)
• 101 offset images of 1/4000 second at ISO 400
3
Image pretreated after integration
Example of a light image
(*) In order to reduce light pollution, an Optolong L-Pro filter has been integrated into the optical path.

4. Agenda
▪ Sensor modeling
▪ Photometry
▪ Appendix
▪ Bibliography
4

5. Reminder of digital sensor specificities (1/2)
5
Photons
Photons strike the
photosite
Some electrons are generated,
they fall into the well
1
2
Measure
At the end of the exposure
time, the electrical voltage is
measured and the ADC
converts it into a digital value
(ADU)
3
ADC(*)
(*) Analog / digital converter
214 − 1
0
11 000
11 000 ADU Display
The numerical values ​​of each
photosite are saved in a file
4
▪ Photoelectron generation process and digital encoding:

6. Reminder of digital sensor specificities (2/2)
▪ The term ADU (Analog to Digital Units) corresponds to the digital value
produced by the ADC (Analog to Digital Converter) from the electrical
voltage of the device.
▪ The ADUs are encrypted on a scale which depends on the dynamic range of
the sensor expressed in bits. In the case study, the sensor encodes in 14
bits, the ADUs therefore take (integer) values ​​between 0 and 214 − 1.
▪ The ADU value is deduced from the number of electrons generated from a
gain parameter denoted by 𝑔𝐴𝐷𝑈/𝑒−. Conversely, when we know the ADU
value of the pixel, we estimate the number of electrons generated with a
gain parameter denoted by 𝑔𝑒−/𝐴𝐷𝑈. We have: 𝑔𝐴𝐷𝑈/𝑒− = 1/𝑔𝑒−/𝐴𝐷𝑈
▪ The electron capacity (full well capacity) represents the maximum number
of electrons that can be stored in the well. The smaller this quantity, the
faster the image saturates.
▪ The Quantum Efficiency is the ratio between the number of electrons
generated and the quantity of photons reaching the sensor.
6
Illustration
Gain shape of EOS 500D

7. Signals and noises
▪ The signals recorded in the image capture process are not deterministic but variables (presence of
noise around the expected value).
▪ There are different types of signals (measured in ADU):
• The signal 𝑆𝑜𝑏𝑗
𝐴𝐷𝑈
(𝑡) of target object,
• The signal 𝑆𝐿𝑃
𝐴𝐷𝑈
(𝑡) related to light pollution (sky background luminosity),
• The Dark Current (DC) signal 𝑆𝐷𝐶
𝐴𝐷𝑈
(𝑡) of the sensor,
• The bias signal (offset) 𝑆𝑅𝑁
𝐴𝐷𝑈
related to the presence of an offset between the pure black level
reference and the non-zero value returned by the sensor. The noise component associated to this
signal is called Read Noise (RN).
▪ Denoting 𝑆𝑡𝑜𝑡
𝐴𝐷𝑈
𝑡 the total signal, we therefore have: 𝑆𝑡𝑜𝑡
𝐴𝐷𝑈
𝑡 = 𝑆𝑜𝑏𝑗
𝐴𝐷𝑈
(𝑡) + 𝑆𝐿𝑃
𝐴𝐷𝑈
(𝑡) + 𝑆𝐷𝐶
𝐴𝐷𝑈
(𝑡) + 𝑆𝑅𝑁
𝐴𝐷𝑈
▪ Under the same notations for the standard deviations, and assuming the independence of the
random sources, it comes:
𝜎𝑡𝑜𝑡
𝐴𝐷𝑈
= 𝜎𝑜𝑏𝑗
𝐴𝐷𝑈
(𝑡)2 + 𝜎𝐿𝑃
𝐴𝐷𝑈
(𝑡)2 + 𝜎𝐷𝐶
𝐴𝐷𝑈
(𝑡)2 + 𝜎𝑅𝑁
𝐴𝐷𝑈
²
7

8. Link between electrons numbers and ADU measures
▪ The signals introduced previously can also be measured in number of electrons (e-). For example consider
the signal (in e-) of the target object, denoted by 𝑆𝑜𝑏𝑗
𝑒−
𝑡 . We therefore have the relation:
𝑆𝑜𝑏𝑗
𝑒−
𝑡 = 𝑔𝑒−/𝐴𝐷𝑈 × 𝑆𝑜𝑏𝑗
𝐴𝐷𝑈
𝑡
▪ It is possible to prove under relevant assumptions (cf. appendix) that for the signals 𝑂𝑏𝑗 and 𝐿𝑃, the
expectation and the variance of the number of electrons generated are equal and linear in 𝒕, the exposure
time.
▪ We write: 𝐸 𝑆𝑋
𝑒−
𝑡 = 𝑉 𝑆𝑋
𝑒−
𝑡 = 𝜆𝑋
𝑒−
. 𝑡 and 𝑋 = 𝑂𝑏𝑗, 𝐿𝑃
▪ From previous equation, it is straightforward to deduce that: 𝑔𝑒−/𝐴𝐷𝑈 =
𝐸 𝑆𝑋
𝐴𝐷𝑈
𝑡
𝑉 𝑆𝑋
𝐴𝐷𝑈 𝑡
▪ Under the same assumptions we prove that the terms 𝑬 𝑺𝑿
𝑨𝑫𝑼
𝒕 𝑎𝑛𝑑 𝑽 𝑺𝑿
𝑨𝑫𝑼
𝒕 are linear in 𝒕. We have:
𝐸 𝑆𝑋
𝐴𝐷𝑈
𝑡 =
𝜆𝑋
𝑒−
𝑔𝑒−/𝐴𝐷𝑈
× 𝑡 𝑎𝑛𝑑 𝑉 𝑆𝑋
𝐴𝐷𝑈
𝑡 =
𝜆𝑋
𝑒−
𝑔𝑒−/𝐴𝐷𝑈
2
× 𝑡
Remark: variance 𝑉 𝑆𝐷𝐶
𝐴𝐷𝑈
𝑡 is also assumed to be linear in 𝑡.
8
Instantaneous rate
related to signal 𝑋

9. Estimation on light, dark and offset images (1/6)
▪ The estimation of different signals expectations and standard
deviations was performed by the process Statistics of PixInsight
software.
▪ For the offset signal, the standard deviation was evaluated on 2
images:
𝜎𝑅𝑁
𝐴𝐷𝑈
=
1
2
𝜎 𝐼𝑜𝑓𝑓𝑠𝑒𝑡
1
− 𝐼𝑜𝑓𝑓𝑠𝑒𝑡
2
+ 𝐶
• Where 𝐶 is a translation factor to avoid truncating negative data
▪ Estimation of offset signal expectation and standard deviation:
9
Offset signal Channel R Channel G Channel B
𝜇𝑅𝑁
𝐴𝐷𝑈 13.7 9.9 11.5
𝜎𝑅𝑁
𝐴𝐷𝑈 10.5 6.6 8.3

10. Estimation on light, dark and offset images (2/6)
▪ The dark signal implicitly integrates the offset signal. To estimate its expectation and standard
deviation, the following adjustments are made:
𝜎𝐷𝐶
𝐴𝐷𝑈
=
1
2
𝑉 𝐼𝑑𝑎𝑟𝑘
1
− 𝐼𝑑𝑎𝑟𝑘
2
+ 𝐷 − 𝜎𝑅𝑁
𝐴𝐷𝑈2
Et
𝜇𝐷𝐶
𝐴𝐷𝑈
= 𝜇 𝐼𝑑𝑎𝑟𝑘
𝑘
− 𝜇𝑅𝑁
𝐴𝐷𝑈
, 𝑘 = 1 𝑜𝑢 2
▪ Estimation of dark signal expectation and standard deviation:
10
Dark signal Channel R Channel G Channel B
𝜇𝐷𝐶
𝐴𝐷𝑈 18.6 13 15.2
𝜎𝐷𝐶
𝐴𝐷𝑈 10.7 8.3 9.7

11. Estimation on light, dark and offset images (3/6)
▪ We also want to perform an estimation of the average signal 𝜇𝐿𝑃
𝐴𝐷𝑈
and the sky background (SB) noise 𝜎𝐿𝑃
𝐴𝐷𝑈
▪ The SB signal integrates the dark and offset signals. To estimate its
expectation and standard deviation, the following adjustments
are made:
𝜎𝐿𝑃
𝐴𝐷𝑈
= 𝜎𝐴
𝐴𝐷𝑈2
− 𝜎𝐷𝐶
𝐴𝐷𝑈2
− 𝜎𝑅𝑁
𝐴𝐷𝑈2
And
𝜇𝐿𝑃
𝐴𝐷𝑈
= 𝜇 𝐼𝑙𝑖𝑔ℎ𝑡ȁ𝐴
− 𝜇𝐷𝐶
𝐴𝐷𝑈
− 𝜇𝑅𝑁
𝐴𝐷𝑈
▪ The term 𝜇 𝐼𝑙𝑖𝑔ℎ𝑡ȁ𝐴
corresponds to the mean value of the signal
measured on an area of ​​SB in the light frame considered. This is a
measure of light pollution (LP):
11
SB signal Channel R Channel G Channel B
𝜇𝐿𝑃
𝐴𝐷𝑈
1694.6 1706.3 1763.6
𝜎𝐿𝑃
𝐴𝐷𝑈 67.9 57.5 65.4

12. Estimation on light, dark and offset images (4/6)
▪ It is possible to estimate the signal of the target object. We
favor a peak luminosity rather than an average luminosity
because it constitutes a more relevant criterion for the
evaluation of the signal / noise ratio.
▪ To estimate the peak luminosity, we select the brightest
area of ​​the object (i.e. the core of the galaxy) which we
reprocess by removing the signal from the central star
(denoted by 𝜇∗
𝐴𝐷𝑈) as well as the other signals (light
pollution, RN and DC):
𝜇𝑂𝑏𝑗
𝐴𝐷𝑈
= 𝜇 𝐼𝑙𝑖𝑔ℎ𝑡ȁ𝑝𝑖𝑐
− 𝜇∗
𝐴𝐷𝑈
− 𝜇𝐷𝐶
𝐴𝐷𝑈
− 𝜇𝑅𝑁
𝐴𝐷𝑈
− 𝜇𝐿𝑃
𝐴𝐷𝑈
▪ Below the estimations of peak luminosities:
12
Signal of Object Channel R Channel G Channel B
𝜇𝑂𝑏𝑗
𝐴𝐷𝑈
342.9 354.5 379.2

13. Estimation on light, dark and offset images (5/6)
▪ For sources 𝑌 = 𝐿𝑃 𝑜𝑟 𝑂𝑏𝑗, we can deduce the expectation of
number of photons 𝜇𝑌
𝑝ℎ
which reach the sensor, by using ADU
measurements carried out:
𝜇𝑌
𝑝ℎ
=
1
𝑄𝐸
×
𝜇𝑌
𝐴𝐷𝑈
𝑡𝑟𝑒𝑓
× 𝑔𝑒−/𝐴𝐷𝑈 ×
1
𝑅 × 𝑅𝐵𝑎𝑦𝑒𝑟
𝑤𝑖𝑡ℎ 𝑌 = 𝐿𝑃 𝑜𝑟 𝑂𝑏𝑗
▪ Under the notations: 𝑄𝐸 = 38% the quantum efficiency, 𝑡𝑟𝑒𝑓 =
240𝑠 the unitary exposure time, 𝑅 = 73% the transmission rate of
the L-Pro filter, 𝑅𝐵𝑎𝑦𝑒𝑟 = 1/3 the transmission rate related to the
matrix of Bayer, 𝑔𝑒−/𝐴𝐷𝑈 = 0.44 the gain in e-/ADU
▪ The average numbers of photons per second and per photosite are
as follows:
13
Photon
Probability of generating a
photoelectron ≡ quantum efficiency
𝑸𝑬 = 𝟑𝟖%
Number of
photons
Channel R Channel G Channel B
𝜇𝐿𝑃
𝑝ℎ
34.0 34.3 35.4
𝜇𝑂𝑏𝑗
𝑝ℎ
6.9 7.1 7.6

14. Estimation on light, dark and offset images (6/6)
▪ In order to validate the Poissonian nature of the photonic signal of the Sky Background, we want to
verify the aforementioned equation:
𝑔𝑒−/𝐴𝐷𝑈 =
𝐸 𝑆𝐿𝑃
𝐴𝐷𝑈
𝑡
𝑉 𝑆𝐿𝑃
𝐴𝐷𝑈
𝑡
▪ Under the notations and adjustments previously introduced, it consists in proving that the term ෡
𝑮
below is equivalent to the gain 𝑔𝑒−/𝐴𝐷𝑈 = 0.44:
෠
𝐺 =
𝜇 𝐼𝑏𝑟𝑢𝑡𝑒ȁ𝐴
− 𝜇𝐷𝐶
𝐴𝐷𝑈
− 𝜇𝑅𝑁
𝐴𝐷𝑈
𝜎𝐴
𝐴𝐷𝑈2
− 𝜎𝐷𝐶
𝐴𝐷𝑈2
− 𝜎𝑅𝑁
𝐴𝐷𝑈2
▪ The results obtained below confirm the Poissonian hypothesis:
14
Gain
parameter
Channel R Channel G Channel B
෠
𝐺 0.37 0.52 0.41
Average 0.43
𝑔𝑒−/𝐴𝐷𝑈 0.44
Result very close to
the target value

15. Signal to Noise Ratio and sub exposure time optimization (1/5)
▪ We seek to optimize the sub exposure time per frame according to criteria associated with the Signal
to Noise ratio (SNR)
▪ We propose to carry out the analysis in several stages:
▪ Reminders on the growth of the SNR according to the number of images;
▪ Sensitivity of the SNR to the sub exposure time;
▪ Optimization of the sub exposure time by Read Noise dominance criterion.
▪ To develop these different notions we will consider simplified SNRs that we will enrich in the rest of
the presentation.
15

16. Signal to Noise Ratio and sub exposure time optimization (2/5)
▪ We recall here the results on the growth of the SNR as a function of the number of images considered.
▪ Below the SNR of a unitary image composed only of the signal of the object and the Read Noise:
𝑆𝑁𝑅1(𝑡) =
𝜆𝑜𝑏𝑗
𝑒−
. 𝑡
𝜆𝑜𝑏𝑗
𝑒−
. 𝑡 + 𝜎𝑅𝑁
𝑒−
²
Where 𝜆𝑜𝑏𝑗
𝑒−
corresponds to the instantaneous rate associated with the signal of the object
▪ Suppose we stack 𝑵 frames of exposure duration 𝑡. So the signal is:
𝑆𝑎 =
1
𝑁
෍
𝑘=1
𝑁
𝑆𝑜𝑏𝑗
𝑒−
𝑘 + 𝑆𝑅𝑁
𝑒−
(𝑘)
▪ We get:
𝑆𝑁𝑅𝑁 𝑁. 𝑡 =
𝐸 𝑆𝑎
𝑉 𝑆𝑎
=
𝜆𝑜𝑏𝑗
𝑒−
. 𝑡
1
𝑁 𝜆𝑜𝑏𝑗
𝑒−
. 𝑡 + 𝜎𝑅𝑁
𝑒−
²
= 𝑁 × 𝑆𝑁𝑅1(𝑡)
16
The SNR increases by a factor 𝑵 compared to the 𝑺𝑵𝑹𝟏 => we exhibit a result well known to
astrophotographers
Assumed centered by
removing the average offset
signal: E 𝑆𝑅𝑁
𝑒−
(𝑘) = 0

17. Signal to Noise Ratio and sub exposure time optimization (3/5)
▪ Let us consider an overall exposure time budget of 𝑇 => this budget can be segmented in different ways.
▪ Consider two sub exposure durations 𝑡1 and 𝑡2 with 𝑡1 < 𝑡2. The associated unitary frame numbers are
denoted 𝑛1 and 𝑛2 and satisfy:
𝑛1. 𝑡1 = 𝑛2. 𝑡2 = 𝑇 𝑎𝑛𝑑 𝑛2 < 𝑛1
▪ Case where the read noise is zero:
𝑆𝑁𝑅𝑛1
(𝑇) = 𝑛1
𝜆𝑜𝑏𝑗
𝑒−
. 𝑡1
𝜆𝑜𝑏𝑗
𝑒−
. 𝑡1
= 𝜆𝑜𝑏𝑗
𝑒−
. 𝑛1. 𝑡1 = 𝜆𝑜𝑏𝑗
𝑒−
. 𝑇 = 𝜆𝑜𝑏𝑗
𝑒−
. 𝑛2. 𝑡2 = 𝑆𝑁𝑅𝑛2
(𝑇)
▪ General case:
𝑆𝑁𝑅𝑛1
𝑇 = 𝑛1
𝜆𝑜𝑏𝑗
𝑒−
. 𝑡1
𝜆𝑜𝑏𝑗
𝑒−
. 𝑡1 + 𝜎𝑅𝑁
𝑒−
²
=
𝜆𝑜𝑏𝑗
𝑒−
. 𝑛1𝑡1
𝜆𝑜𝑏𝑗
𝑒−
. 𝑛1𝑡1 + 𝑛1. 𝜎𝑅𝑁
𝑒−
²
=
𝜆𝑜𝑏𝑗
𝑒−
. 𝑇
𝜆𝑜𝑏𝑗
𝑒−
. 𝑇 + 𝑛1. 𝜎𝑅𝑁
𝑒−
²
<
𝜆𝑜𝑏𝑗
𝑒−
. 𝑇
𝜆𝑜𝑏𝑗
𝑒−
. 𝑇 + 𝑛2. 𝜎𝑅𝑁
𝑒−
²
= 𝑆𝑁𝑅𝑛2
(𝑇)
17
The segmentation strategy has no impact.
Only the total exposure time matters.
Read noise is counted as many times as sub frames are produced. In conclusion, the longest possible
exposures should be made (i.e. 𝑇) even if in practice it is impossible for technical reasons related to the
realization of the frames (tracking errors, occurrences of planes or satellites,... ).

18. Signal to Noise Ratio and sub exposure time optimization (4/5)
▪ We seek here to optimize the sub exposure time by read noise dominance criterion(*). Consider the
expression of the SNR below in which we introduce the noise associated with light pollution:
𝑆𝑁𝑅1 𝑡 =
𝜆𝑜𝑏𝑗
𝑒−
. 𝑡
𝜆𝑜𝑏𝑗
𝑒−
. 𝑡 + 𝜆𝐿𝑃
𝑒−
. 𝑡 + 𝜎𝑅𝑁
𝑒−
²
=
𝜆𝑜𝑏𝑗
𝑒−
. 𝑡
𝜆𝑜𝑏𝑗
𝑒−
. 𝑡 + 𝜆𝐿𝑃
𝑒−
. 𝑡. 1 +
𝜎𝑅𝑁
𝑒−
²
𝜆𝐿𝑃
𝑒−
. 𝑡
▪ The objective is, given a level of tolerance 𝜶, to find the minimum time, denoted 𝑡𝑚𝑖𝑛, such that for 𝑡 ≥
𝑡𝑚𝑖𝑛 we have:
𝜆𝐿𝑃
𝑒−
. 𝑡. 1 +
𝜎𝑅𝑁
𝑒−
²
𝜆𝐿𝑃
𝑒−
. 𝑡
≤ 1 + 𝛼 𝜆𝐿𝑃
𝑒−
. 𝑡
▪ We get:
𝒕𝒎𝒊𝒏 =
𝟏
𝟏 + 𝜶 𝟐 − 𝟏
×
𝝈𝑹𝑵
𝒆− 𝟐
𝝀𝑳𝑷
𝒆−
18
(*) Method implemented by Robin Glover the developer of the SharpCap astrophotography software
For 𝑡 ≥ 𝑡𝑚𝑖𝑛 the residual noise (i.e. in addition to the variance of the object signal) is at most
equal to the noise induced by the light pollution increased by the threshold 𝜶.

19. Signal to Noise Ratio and sub exposure time optimization (5/5)
▪ To estimate 𝑡𝑚𝑖𝑛, we use the ADU measurements
detailed above with the equation:
𝑡𝑚𝑖𝑛 =
1
1 + 𝛼 2 − 1
×
𝜎𝑅𝑁
𝐴𝐷𝑈2
𝜎𝐿𝑃
𝐴𝐷𝑈
𝑡𝑟𝑒𝑓
2
/𝑡𝑟𝑒𝑓
Where 𝑡𝑟𝑒𝑓 represents the reference time of the sub frame
used to estimate 𝜎𝐿𝑃
𝐴𝐷𝑈
(in the case study 𝑡𝑟𝑒𝑓 =240sec).
▪ Below the values ​​associated with the 2% and 5%
thresholds:
19
Threshold 𝑡𝑚𝑖𝑛(𝑅) 𝑡𝑚𝑖𝑛(𝐺) 𝑡𝑚𝑖𝑛(𝐵) 𝑀𝒂𝒙 𝑨𝒗𝒆𝒓𝒂𝒈𝒆
𝜶 = 𝟐% 141 79 95 141 105
𝜶 = 𝟓% 56 31 38 56 41
With a sub exposure
value of 141sec it
possible to satisfy the
dominance criterion
for thresholds ≥2%

20. Dynamics criterion (1/2)
▪ The sky background signal induced by light pollution (which
represents the area of lowest intensity on each image) increases
with exposure time and therefore reduces the range of signal
collected (loss of dynamics range). We propose here a criterion for
determining a 𝑡𝑚𝑎𝑥 allowing to control the loss in dynamics.
▪ The dynamics range of an image in bits is calculated as follows:
𝐷 𝑡 = 𝑙𝑜𝑔2
𝑆𝑚𝑎𝑥
𝑆𝑚𝑖𝑛
20
Where 𝑆𝑚𝑎𝑥 corresponds to the saturation (difference between the maximum value in ADU and the smallest recorded
signal value) and 𝑆𝑚𝑖𝑛 a minimum signal reference (most often we choose 𝑆𝑚𝑖𝑛 = 𝜎𝑅𝑁).
▪ In our case study we calculate the dynamics by the relation:
𝐷 𝑡 = 𝑙𝑜𝑔2
2𝑏 − 𝑡 × 𝑔𝑒−/𝐴𝐷𝑈 × 𝜎𝐿𝑃
𝐴𝐷𝑈
𝑡𝑟𝑒𝑓
2
/𝑡𝑟𝑒𝑓
𝜎𝑅𝑁
▪ Consequence : given a dynamics loss tolerance hypothesis, denoted by ∆𝑫 (in bits), we can then determine
the maximum sub exposure time, denoted by 𝑡𝑚𝑎𝑥, through the image dynamics criterion.
The dynamics decreases with 𝑡

21. Dynamics criterion (2/2)
▪ Thus, we define 𝑡𝑚𝑎𝑥 as follows:
𝐷 𝑡𝑚𝑖𝑛 − 𝐷 𝑡𝑚𝑎𝑥 = ∆𝐷
▪ To calculate 𝑡𝑚𝑎𝑥, we use the equation below:
𝑡𝑚𝑎𝑥 =
2𝑏
− 2𝐷 𝑡𝑚𝑖𝑛 −∆𝐷
× 𝜎𝑅𝑁
𝑔𝑒−/𝐴𝐷𝑈 × 𝜎𝐿𝑃
𝐴𝐷𝑈
𝑡𝑟𝑒𝑓
2
/𝑡𝑟𝑒𝑓
▪ Most often, experts recommend a value of ∆𝑫 close
to 0.8 bits. Below the detail of the 𝑡𝑚𝑎𝑥 estimation:
21
Estimation of 𝒕𝒎𝒂𝒙
𝑡𝑚𝑖𝑛 𝑠 (*) 30
∆𝐷 (𝑏𝑖𝑡𝑠) 0.8
𝒕𝒎𝒂𝒙 (𝒔) 225
(*) The value 𝑡𝑚𝑖𝑛 considered is the
smallest admissible element (31 sec
exactly) among the values ​​obtained
previously

22. Tracking error criterion on sub frames (1/2)
▪ Operationally, the implementation of autoguiding can sometimes - on a small part of the photographs taken -
produce sub frames with anomalies (tracking errors causing star trails).
▪ In this part, we seek to objectify an optimal sub exposure time with the objective to minimize the defection rate of
the sub frames set.
▪ The number of autoguiding incidents is assumed to follow a Poisson process with parameter 𝜆 which can be
estimated from the empirical success rates 𝑺𝑹 =
#𝐼𝑚𝑎𝑔𝑒𝑠 𝑠𝑎𝑛𝑠 𝑑é𝑓𝑎𝑢𝑡
#𝐼𝑚𝑎𝑔𝑒𝑠 𝑡𝑜𝑡𝑎𝑙
▪ Below a table summarizing the 𝑆𝑅 of different sessions:
22
Sample NGC869 IC405 IC1805 M33 M31 M31 M31 M31 NGC281 NGC6888 NGC7023
Name
Double
cluster
Flaming Star
Heart
Nebula
Triangulum Andromeda Andromeda Andromeda Andromeda PacMan Crescent Iris
Time exposure
(s)
60 180 240 240 180 180 180 180 300 180 240
#Selected
frames
108 76 53 80 36 25 67 44 49 135 72
#Non selected
frames
4 6 9 9 4 7 5 5 1 14 8
#Total frames 112 82 62 89 40 32 72 49 50 149 80
Success rate 96% 93% 85% 90% 90% 78% 93% 90% 98% 91% 90%

23. Tracking error criterion on sub frames (2/2)
▪ The estimator መ
𝜆 of 𝜆 is defined as the empirical mean of the
estimators per session:
መ
𝜆 = −
1
𝑀
෍
𝑘=1
𝑀
1
𝑡𝑘
𝑙𝑛 𝑇𝑆𝑘
▪ Below the results of estimation:
▪ By denoting 𝛼 the maximum tolerated defection rate, we deduce the
maximum sub exposure time, denoted 𝑡𝑑𝑒𝑓, satisfying the criterion
𝑇𝑒−𝜆𝑡𝑑𝑒𝑓 ≥ 𝑇 1 − 𝛼 . This time is estimated by using the formula:
𝑡𝑑𝑒𝑓 = −
1
𝜆
𝑙𝑛 1 − 𝛼
▪ For 𝛼 = 10%, we have 𝑡𝑑𝑒𝑓 = −
1
0,056%
𝑙𝑛 1 − 10% , and
𝒕𝒅𝒆𝒇 = 𝟏𝟖𝟖 𝒔𝒆𝒄
23
Lambda estimations
Average 0.056%
Median 0.055%
For greater caution,
we retain the value
෠
𝝀 = 0.056%

24. Agenda
▪ Sensor modeling
▪ Photometry
▪ Appendix
▪ Bibliography
24

25. Spectral band, energy et photons flux (1/2)
▪ We focus in the present study on the spectral bands related to 3
typical domains (Blue, Visible and Red)
▪ For each of these bands, a flux density (in W/m²/nm) expressed
in the Vega reference system allows to evaluate a flux of
photons for each pixel of the sensor.
▪ We recall the formula for calculating the photon energy:
𝐸 = ℎ
𝑐
𝜆
With ℎ the Planck constant equal to 6.6262. 10−34
𝐽. 𝑠 and 𝑐 the speed of light
equal to 2.9979. 108
𝑚/𝑠
▪ The following table details the properties of the considered
spectral bands:
25
Let 𝑤(𝜆) 𝜆 be the transmission
rates of a spectral band, then the
bandwidth is the quantity 𝑩𝑾 ≜
‫׬‬𝝀𝒎𝒊𝒏
𝝀𝒎𝒂𝒙
𝒘(𝝀)𝒅𝝀 and the average
wavelength is calculated by using the
formula
𝟏
𝑩𝑾
‫׬‬𝝀𝒎𝒊𝒏
𝝀𝒎𝒂𝒙
𝝀. 𝒘(𝝀)𝒅𝝀.
Spectral band
Flux density W /
m²/ nm
Average wavelength
(nm)
Bandwidth (nm) Photon energy (J)
Flux of photons
(photon / s / cm²)
B 6.60085E-11 422 66 4.70489E-19 926 048
V 3.60994E-11 549 102 3.62E-19 1 017 951
R 2.28665E-11 662 141 3.0003E-19 1 075 977

26. Spectral band, energy et photons flux (2/2)
▪ In the following, for all 3 spectral domains, we will only refer to the flux density 𝑭𝑽 associated with the
visible domain, and we will consider a single bandwidth.
▪ The retained bandwidth (denoted by ∆) corresponds to the widths of the unitary spectral bands
(denoted by 𝐿𝐵𝑆𝐵, 𝐿𝐵𝑆𝑉 𝑎𝑛𝑑 𝐿𝐵𝑆𝑅) weighted by flux densities (denoted by 𝐹𝐵, 𝐹𝑉 𝑎𝑛𝑑 𝐹𝑅). We get:
∆= 𝐿𝐵𝑆𝐵 ×
𝐹𝐵
𝐹𝑉
+ 𝐿𝐵𝑆𝑉 + 𝐿𝐵𝑆𝑅 ×
𝐹𝑅
𝐹𝑉
▪ From the assumptions considered, we have:
∆≈ 66 ×
6.60085. 10−11
3.60994. 10−11
+ 102 + 141 ×
2.28665. 10−11
3.60994. 10−11
≈ 312 𝑛𝑚.
▪ We will retain in the following the parameterization: ∆≈ 𝟑𝟎𝟎 𝒏𝒎.
26

27. Link between flux and magnitude
▪ We recall below the formula to evaluate the flux 𝑭𝒎 of a sky object according to its magnitude 𝒎
expressed in the Vega system (zero magnitude for Vega):
𝑭𝒎 = 𝑭𝑽 × 𝟏𝟎−𝒎/𝟐.𝟓
With 𝐹𝑉 = 3.60994. 10−11 𝑊/𝑚2/𝑛𝑚.
▪ The elements 𝐹𝑚 and 𝐹𝑉 above correspond to point source flux which can be adjusted in surface
flux expressed for example in 𝑊/𝑚2
/𝑛𝑚 /𝑎𝑟𝑐 𝑠𝑒𝑐² without modifying the calculation formula.
27

28. Photon counting (1/3)
▪ Let us introduce an optical framework composed of a telescope and a digital sensor. Let 𝐹 (resp. 𝐷) be the focal
length in 𝑚𝑚 (resp. the diameter in 𝑚𝑚) of the telescope and 𝑃 the size of a sensor pixel (in 𝜇𝑚).
▪ The number of photons, denoted by 𝑵𝒑, from an extended source of magnitude 𝒎 and solid angle 𝛀 expressed
in 𝑎𝑟𝑐 𝑠𝑒𝑐², for each pixel per second can be estimated using the formula:
𝑵𝒑 = 𝟏𝟎−𝒎/𝟐.𝟓 ×
∆. 𝑭𝑽
𝑬
×
𝝅. 𝑫𝟐
𝟒
× 𝛀
▪ The following formula evaluates the number of arc sec / pixel: 𝐸 (𝑎𝑟𝑐 𝑠𝑒𝑐) = 206 ×
𝑃(𝜇𝑚)
𝐹(𝑚𝑚)
▪ We consider the approximation: Ω(𝑎𝑟𝑐 𝑠𝑒𝑐²) = 𝐸2, thus:
𝑁𝑝 = 10−𝑚/2.5
×
∆. 𝐹𝑉
𝐸
×
𝜋. 𝐷2
4
× 206 ×
𝑃(𝜇𝑚)
𝐹(𝑚𝑚)
2
= 10−𝑚/2.5
×
∆. 𝐹𝑉
𝐸
×
𝜋. 2062
4
×
𝑃(𝜇𝑚)
𝑓
2
▪ We get a synthetic equation for calculating the number of photons:
𝑵𝒑 = 𝑪 × 𝟏𝟎−𝒎/𝟐.𝟓 ×
𝑷(𝝁𝒎)
𝒇
𝟐
Under assumptions made, we have: 𝐶 ≈ 9.98. 108
28
𝑓 ≜
𝐹
𝐷
is the
telescope aperture
ratio

29. Photon counting (2/3)
▪ We apply the previous formula to the Sky Background signal and to the object signal.
▪ Sky Background Brightness
▪ Suppose that the observation location where the image is taken has an SQM (Sky Quality Meter)
equal to 18.16 𝑚𝑎𝑔 / arcsec². The flux of photons per second associated with the Background
Sky (BS) signal 𝜇𝐿𝑃
𝑝ℎ
is as follows:
𝜇𝐿𝑃
𝑝ℎ
= 9.98. 108 × 10−18.16/2.5 ×
4.68
5
2
= 𝟒𝟕. 𝟔𝟏 𝒑𝒉𝒐𝒕𝒐𝒏𝒔 / 𝒔
▪ Object brightness
▪ The surface brightness of the target object (galaxy M33) is 23.25 𝑚𝑎𝑔 / arcsec². We
nevertheless prefer the peak luminosity(*) equal to 20.1 𝑚𝑎𝑔 / 𝑎𝑟𝑐 sec² for the calculation of
the signal of the object. The flux of photons per second associated with the signal from the
object 𝜇𝑂𝑏𝑗
𝑝ℎ
is as follows:
𝜇𝑂𝑏𝑗
𝑝ℎ
= 9.98. 108 × 10−20.1/2.5 ×
4.68
5
2
= 𝟕. 𝟗𝟕 𝒑𝒉𝒐𝒕𝒐𝒏𝒔 / 𝒔
29
(*) See the Messier catalog proposed by Tony Flanders: https://tonyflanders.wordpress.com/messier-guide-index-by-number/

30. Photon counting (3/3)
▪ The table below shows the empirical estimates of the number of photons per channel as well as the
theoretical estimates:
▪ The empirical and theoretical photon numbers associated with the signal from the object are very
close.
▪ Nevertheless, there is a discrepancy in the numbers of photons of the Sky Background signal which
is largely due to the SQM hypothesis. This assumption was not evaluated for the observation spot
but was extracted from the Light Pollution Map(*) site.
▪ As an example, an SQM of 18.5 𝑚𝑎𝑔/arcsec² leads to an average number of 34.81 photons/s much
closer to the calculated empirical values. The sensitivity to this assumption is therefore particularly
substantial.
30
(*) https://www.lightpollutionmap.info/
Number of
photons
Channel R Channel G Channel B
Theoretical
calculation
𝜇𝐿𝑃
𝑝ℎ
34.0 34.3 35.4 47.61
𝜇𝑂𝑏𝑗
𝑝ℎ
6.9 7.1 7.6 7.97

31. SNR value and profile (1/2)
▪ It is possible to exploit the previous photometry results to evaluate the signal-to-noise ratio of the
observation site 𝐻 (for Home) where the image was made.
▪ By denoting 𝑡𝐻 the total exposure time, 𝑡𝑢 the unit exposure time, the signal-to-noise ratio is equal to:
𝑆𝑁𝑅𝑛𝐻
(𝑡𝐻) =
𝜆𝑜𝑏𝑗
𝑒−
𝑔𝑒−/𝐴𝐷𝑈
𝜆𝑜𝑏𝑗
𝑒−
𝑔𝑒−/𝐴𝐷𝑈
2 +
𝜎𝐿𝑃
𝐴𝐷𝑈
𝑡𝑟𝑒𝑓
2
𝑡𝑟𝑒𝑓
+
𝜎𝐷𝐶
𝐴𝐷𝑈
𝑡𝑟𝑒𝑓
2
𝑡𝑟𝑒𝑓
+
𝜎𝑅𝑁
𝐴𝐷𝑈
²
𝑡𝑢
× 𝑡𝐻
With: 𝜆𝑜𝑏𝑗
𝑒−
= 𝑄𝐸 × 𝑅 × 𝑅𝐵𝑎𝑦𝑒𝑟 × 𝐶 × 10−𝑚𝑂𝑏𝑗/2.5
×
𝑃(𝜇𝑚)
𝑓
2
31
For 𝑡𝑢 = 240𝑠, we have the following
signal-to-noise ratio : 𝑺𝑵𝑹=𝟒𝟔.𝟗
Remember that the total exposure time of
the case study is 5h20’

32. SNR value and profile (2/2)
▪ By setting a target SNR denoted 𝑆𝑁𝑅𝑡𝑔𝑡
, the total exposure time 𝑡𝑡𝑔𝑡
required is calculated as follows:
𝑡𝑡𝑔𝑡 = 𝐶𝐻
𝑆𝑁𝑅𝑡𝑔𝑡. 𝑔𝑒−/𝐴𝐷𝑈
𝜆𝑜𝑏𝑗
𝑒−
2
Where,
𝐶𝐻 =
𝜆𝑜𝑏𝑗
𝑒−
𝑔𝑒−/𝐴𝐷𝑈
2
+
𝜎𝐿𝑃
𝐴𝐷𝑈
𝑡𝑟𝑒𝑓
2
𝑡𝑟𝑒𝑓
+
𝜎𝐷𝐶
𝐴𝐷𝑈
𝑡𝑟𝑒𝑓
2
𝑡𝑟𝑒𝑓
+
𝜎𝑅𝑁
𝐴𝐷𝑈
²
𝑡𝑢
32
Astrophotography experts often consider a target of 𝑺𝑵𝑹 = 𝟓𝟎.
To achieve such an SNR, the exposure time must be 6h07' (instead of 5h20' in the case study).

33. Alternative approach of Read Noise dominance
▪ This alternative technique aims to improve the
previously detailed Read Noise dominance method,
by considering the entire signal-to-noise ratio as the
basis for calculation.
▪ We introduce 𝑉 =
𝜆𝑜𝑏𝑗
𝑒−
𝑔𝑒−/𝐴𝐷𝑈
2 + 𝜎𝐿𝑃
𝐴𝐷𝑈
𝑡𝑟𝑒𝑓
2
×
1
𝑡𝑟𝑒𝑓
+
𝜎𝐷𝐶
𝐴𝐷𝑈
𝑡𝑟𝑒𝑓
2
×
1
𝑡𝑟𝑒𝑓
▪ We solve the following inequality:
𝜆𝑜𝑏𝑗
𝑒−
. 𝑡/𝑔𝑒−/𝐴𝐷𝑈
𝑡𝑉 + 𝜎𝑅𝑁
𝐴𝐷𝑈
²
≥ 1 + 𝛼
𝜆𝑜𝑏𝑗
𝑒−
. 𝑡/𝑔𝑒−/𝐴𝐷𝑈
𝑡𝑉
▪ Solving the inequality leads to the following
solution 𝒕𝒎𝒊𝒏
∗
:
𝑡𝑚𝑖𝑛
∗
=
𝜎𝑅𝑁
𝐴𝐷𝑈2
𝑉
×
1
1 + 𝛼 2 − 1
33
Threshold Alternative
method
Initial method
𝜶 = 𝟐% 116 141
𝜶 = 𝟓% 46 56
Values of 𝑡𝑚𝑖𝑛 in sec (channel R)
SNR excluding
RN

34. Summary of sub exposure time constraints (1/2)
34
Maximum sub
exposure time
Minimum sub
exposure time
Minimum exposure time
associated with the Read Noise
dominance (alternative method
𝛼 = 5%)
𝒕𝒎𝒊𝒏 = 𝟒𝟔 𝒔
Minimum exposure time associated
with the Read Noise dominance
(alternative method 𝛼 = 2%)
𝒕𝒎𝒊𝒏 = 𝟏𝟏𝟔 𝒔
Minimum exposure time
associated with the Read Noise
dominance (standard method
𝛼 = 2%)
𝒕𝒎𝒊𝒏 = 𝟏𝟒𝟏 𝒔
Maximum exposure
time associated with the
defection rate criterion
𝒕𝒅𝒆𝒇 = 𝟏𝟖𝟖 𝒔
Maximum exposure time
associated with the loss in
dynamics criterion
𝒕𝒅𝒚𝒏 = 𝟐𝟐𝟓 𝒔
Recommended exposure
time satisfying all the
criteria
𝒕𝒖 = 𝟏𝟓𝟎 𝒔
Sub exposure time of
the case study
𝒕𝒄𝒂𝒔𝒆 𝒔𝒕𝒖𝒅𝒚 = 𝟐𝟒𝟎 𝒔

35. Summary of sub exposure time constraints (2/2)
35
▪ The graph below displays the constraints induced by the dominance method, the management of the
defection rate and of the dynamics:
Range of
admissible sub
exposure times

36. Comparison of 2 observation sites SNRs (1/3)
▪ The practitioner may need to compare the total exposure time between an area where light pollution
is intense and a spot with a relatively pure sky background.
▪ We can for example ask the following question => What is the required exposure time (denoted 𝑡𝐻)
on an area of strong light pollution (called 𝐻 for Home) to obtain the same SNR as that resulting from
a fixed exposure duration (denoted 𝑡𝑂𝑆) on a low polluted site (called 𝑂𝑆 for Other Site)?
▪ Let’s introduce the SNRs of each of the sites:
𝑆𝑁𝑅𝑛𝐻
𝑡𝐻 =
𝜆𝑜𝑏𝑗
𝑒−
𝑔𝑒−/𝐴𝐷𝑈 𝐶𝐻
× 𝑡𝐻 𝑤𝑖𝑡ℎ 𝐶𝐻 =
𝜆𝑜𝑏𝑗
𝑒−
𝑔𝑒−/𝐴𝐷𝑈
2 +
𝜎𝐿𝑃
𝐴𝐷𝑈
𝑡𝑟𝑒𝑓
2
𝑡𝑟𝑒𝑓
+
𝜎𝐷𝐶
𝐴𝐷𝑈
𝑡𝑟𝑒𝑓
2
𝑡𝑟𝑒𝑓
+
𝜎𝑅𝑁
𝐴𝐷𝑈
²
𝑡𝑢
And
𝑆𝑁𝑅𝑛𝑂𝑆
𝑡𝑂𝑆 =
𝜆𝑜𝑏𝑗
𝑒−
𝑔𝑒−/𝐴𝐷𝑈 𝐶𝑂𝑆
𝑡𝑂𝑆 𝑤𝑖𝑡ℎ 𝐶𝑂𝑆 =
𝜆𝑜𝑏𝑗
𝑒−
𝑔𝑒−/𝐴𝐷𝑈
2 +
𝑂𝑆
𝜆𝐿𝑃
𝑒−
𝑔𝑒−/𝐴𝐷𝑈
2 +
𝜎𝐷𝐶
𝐴𝐷𝑈
𝑡𝑟𝑒𝑓
2
𝑡𝑟𝑒𝑓
+
𝜎𝑅𝑁
𝐴𝐷𝑈
²
𝑡𝑢
With 𝑛𝐻 =
𝑡𝐻
𝑡𝑢
and 𝑛𝑂𝑆 =
𝑡𝑂𝑆
𝑡𝑢
36

37. Comparison of 2 observation sites SNRs (2/3)
▪ In the previous expressions the signal of the object is
estimated by means of the relation:
𝜆𝑜𝑏𝑗
𝑒−
= 𝑄𝐸 × 𝑅 × 𝑅𝐵𝑎𝑦𝑒𝑟 × 𝐶 × 10−𝑚𝑂𝑏𝑗/2,5
×
𝑃(𝜇𝑚)
𝑓
2
▪ Furthermore, it is assumed that the noise of sky
background (SB) is empirically calculated for 𝐻 (on a
light frame) and is theoretically estimated for 𝑂𝑆 from an
index 𝑺𝑸𝑴𝑶𝑺 = 𝟐𝟏. 𝟐𝟑 mag / arc sec² with the formula:
𝑂𝑆𝜆𝐿𝑃
𝑒−
= 𝑄𝐸 × 𝑅 × 𝑅𝐵𝑎𝑦𝑒𝑟 × 𝐶 × 10−𝑆𝑄𝑀𝑂𝑆/2.5 ×
𝑃(𝜇𝑚)
𝑓
2
▪ We get the following equation of time:
𝑡𝐻
∗
=
𝐶𝐻
𝐶𝑂𝑆
× 𝑡𝑂𝑆
And the final result:
37
Intermediate elements Value
Signal (ADU / s) : 𝜆𝑜𝑏𝑗
𝑒−
/𝑔𝑒−/𝐴𝐷𝑈 1.7
Variance Signal (ADU / s) : 𝜆𝑜𝑏𝑗
𝑒−
/
𝑔𝑒−/𝐴𝐷𝑈
2 3.7
Variance SB (ADU / s) for 𝐻 :
𝜎𝐿𝑃
𝐴𝐷𝑈
𝑡𝑟𝑒𝑓
2
/𝑡𝑟𝑒𝑓
19.2
Variance DC (ADU / s) : 𝜎𝐷𝐶
𝐴𝐷𝑈
𝑡𝑟𝑒𝑓
2
/
𝑡𝑟𝑒𝑓
0.48
Variance RN (ADU / frame) : 𝜎𝑅𝑁
𝐴𝐷𝑈
² 110
Variance SB (ADU / s) for 𝑂𝑆 : 𝑂𝑆
𝜆𝐿𝑃
𝑒−
. 𝑡𝑂𝑆/
𝑔𝑒−/𝐴𝐷𝑈
2 1.3
Sub exposure time (s) : 𝑡𝑢 150
Time ratio Value
Total variance (ADU / s) for 𝐻 : 𝐶𝐻 24
Total variance (ADU / s) for 𝑂𝑆 : 𝐶𝑂𝑆 6
Total exposure time for 𝑂𝑆 (s) : 𝑡𝑂𝑆 1
Total exposure time for 𝐻 (s) : 𝑡𝐻
∗
3.86
𝒕𝑯
∗
≈ 𝟑. 𝟖𝟔 × 𝒕𝑶𝑺

38. Comparison of 2 observation sites SNRs (3/3)
Comparison of times to reach a
target SNR:
38
Time required to achieve an SNR = 50
Site 𝑂𝑆: 𝑡𝑂𝑆 1h35’
Site 𝐻: 𝑡𝐻 6h07’
From an operational point of view, these orders of magnitude induce very different constraints. For
example, in the case of an astronomical night whose duration is less than 5h, it is impossible to
obtain an SNR ratio of 50 for the site 𝑯 even though the latter is reached in 1h35 for 𝑶𝑺.

39. Sensitivity of the SNR to the signal of object value
39
▪ The calculations presented above derive from a theoretical
assumption of so-called "peak" luminosity equal to 20.1
mag/arcsec². This assumption leads to a flux of 7.97
photons/s, emanating from the brightest part of the target
object.
▪ In addition, we have also seen that the empirical
measurement of the signal in the core of M33 directly
carried out on a sub frame, enabled us to estimate a value
of 7.21 photons/s (average of the R, G and B channels).
▪ We present here a sensitivity analysis of the SNR to this
estimate.
Intermediate elements
Signal (ADU / s) - initial value:
𝜆𝑜𝑏𝑗
𝑒−
/𝑔𝑒−/𝐴𝐷𝑈
1.7
Variance Signal (ADU / s) - initial
value: 𝜆𝑜𝑏𝑗
𝑒−
/𝑔𝑒−/𝐴𝐷𝑈
2 3.7
Signal (ADU / s) - empirical value:
𝑎𝑙𝑡
𝜆𝑜𝑏𝑗
𝑒−
/𝑔𝑒−/𝐴𝐷𝑈
1.5
Variance Signal (ADU / s) - empirical
value: 𝑎𝑙𝑡
𝜆𝑜𝑏𝑗
𝑒−
/𝑔𝑒−/𝐴𝐷𝑈
2 3.4
Time required to achieve an SNR = 50
𝑡𝐻 initial value 6h07’
𝑡𝐻 empirical value 7h20’

40. Agenda
▪ Sensor modeling
▪ Photometry
▪ Appendix
▪ Bibliography
40

41. Photons number law of probability: introduction
▪ It is most often assumed that the number of photons emitted by astronomical objects or the signal from
the Sky Background follows a Poisson process. In general, two arguments are developed to justify this
phenomenon.
▪ The first consists in considering a set of hypotheses similar to the mathematical definition of the Poisson
process. Below, these assumptions:
• The probability of detecting a photon over an infinitesimal time interval is proportional to an intensity parameter:
𝑃 𝑁𝑡+∆𝑡 − 𝑁𝑡 = 1 = 𝜆∆𝑡 + 𝑜 ∆𝑡 ;
• The probability of detecting a quantity greater than one photon is negligible: 𝑃 𝑁𝑡+∆𝑡 − 𝑁𝑡 > 1 = 𝑜 ∆𝑡 ;
• Photon detections associated with disjoint time intervals are statistically independent.
▪ Under these assumptions, the number of photons emitted by the target object follows a Poisson process.
In mathematics, a Poisson process is exactly defined by the 3 hypotheses formulated above.
▪ An alternative strategy consists in showing that the poissonian distribution of light comes from intrinsic
properties of electromagnetic radiation. Thus it is possible (under good assumptions) to demonstrate using
quantum mechanics results associated with the quantum harmonic oscillator that the number of photons
is indeed Poisson distributed.
41

42. Photons number law of probability: quantum mechanics (1/3)
▪ In quantum optics, a constant intensity light source can be
modeled as a coherent state of a quantum harmonic
oscillator.
▪ We recall the expression of the quantum Hamiltonian:
𝐻 =
1
2
ℏ𝜔 𝑄2 + 𝑃2
With
𝑄 =
𝑚𝜔
ℏ
ො
𝑞 𝑒𝑡 𝑃 =
1
𝑚𝜔ℏ
Ƹ
𝑝
▪ We define the annihilation and creation operators denoted
𝑎 and 𝑎∗ respectively as follows:
𝑎 =
1
2
𝑄 + 𝑖𝑃 𝑒𝑡 𝑎∗
=
1
2
𝑄 − 𝑖𝑃
▪ The number operator, denoted 𝑁, verifies: 𝑁 = 𝑎∗𝑎
▪ It is easily shown that: 𝐻 = ℏ𝜔 𝑁 +
1
2
42
Theorem:
(i) The spectrum of 𝑁 is ℕ. It follows that the
spectrum of 𝐻 is 𝑛 +
1
2
ℏ𝜔ȁ 𝑛 ∈ ℕ
(ii) If ۧ
ȁ𝜓𝑛 is an eigenvector of 𝑁 associated
with the eigenvalue 𝒏, then 𝑎∗ ۧ
ȁ𝜓𝑛 is an
eigenvector of 𝑁 associated with the
eigenvalue 𝒏 + 𝟏
(iii) If ۧ
ȁ𝜓𝑛 is an eigenvector of 𝑁 associated
with the eigenvalue 𝒏, then if 𝑛 ≠ 0, 𝑎 ۧ
ȁ𝜓𝑛
is an eigenvector of 𝑁 associated with the
eigenvalue 𝒏 − 𝟏 and if 𝑛 = 0, 𝑎 ۧ
ȁ𝜓0 = 0

43. Photons number law of probability: quantum mechanics (2/3)
▪ Note that the eigenstates of 𝑵 and 𝑯 are identical. These states are also called Fock states.
▪ We show that if 𝝍𝒏 is an eigenstate of 𝑵 - or equivalently of 𝐻 - associated with the eigenvalue 𝑛 or
𝐸𝑛 = 𝑛 +
1
2
ℏ𝜔 then:
𝜓𝑛 𝑢 =
1
𝜋
1
4 1
2𝑛𝑛!
𝑒−
𝑢2
2 𝐻𝑛 𝑢
With 𝐻𝑛 𝑢 the Hermite polynomial of order 𝑛.
𝐻0 𝑢 = 1, 𝐻1 𝑢 = 2𝑢, 𝐻𝑛+1 𝑢 = 2𝑢𝐻𝑛 𝑢 − 2𝑛𝐻𝑛−1 𝑢 𝑛 ≥ 2.
▪ Most often we denote ۧ
ȁ𝒏 ≡ 𝝍𝒏, the Fock state associated with the eigenvalue 𝑛 of the operator 𝑁
and the eigenvalue 𝐸𝑛 = 𝑛 +
1
2
ℏ𝜔 of the operator 𝐻.
43
It is assumed that the system is in a coherent state. These states, also called quasi-classical states,
correspond to the purest states that can be defined. In particular, they minimize the uncertainty
principle and come close to the solutions of the classical harmonic oscillator.

44. Photons number law of probability: quantum mechanics (3/3)
▪ Coherent states are defined as the eigenstates of the 𝒂 operator. Let 𝑧 ∈ ℂ, so we have: 𝑎 ۧ
ȁ𝑧 = ۧ
𝑧ȁ𝑧
▪ And we can prove the following formula: ۧ
ȁ𝑧 = σ𝑛
𝑧𝑛
𝑛!
𝑒− 𝑧 2/2 ۧ
ȁ𝑛
▪ If the system is in the coherent state ۧ
ȁ𝒛 , then we can determine the probability associated with the energy
value 𝑬𝒏 = 𝒏 +
𝟏
𝟐
ℏ𝝎. This probability, denoted 𝑷𝒏 𝒛 , corresponds to the squared modulus of the
projection on the Fock state associated with 𝐸𝑛. So we have :
𝑃𝑛 𝑧 =
𝑧𝑛
𝑛!
𝑒− 𝑧 2/2
2
=
𝑧 2 𝑛
𝑛!
𝑒− 𝑧 2
▪ We get the elementary probability of a Poisson distribution (with parameter 𝑧 2) associated with the count
of 𝑛 occurrences.
▪ Under the usual notations of quantum mechanics, we can therefore write that:
𝑁 ۧ
ȁ𝑧 = 𝑧 2
𝑒𝑡 ∆𝑁 ۧ
ȁ𝑧 = 𝑧 2 = 𝑧
44
We conclude that the number of photons measured from a coherent state follows a Poisson
distribution. An immediate consequence is that, over a fixed time interval, the expectation and the
variance of the number of observed photons are identical.

45. Equality expectation / variance of the photoelectrons number (1/2)
▪ The number of photons reaching the sensor is therefore assumed to follow a Poisson distribution. As an
example, consider 𝑁𝑜𝑏𝑗
𝑝ℎ
𝑡 the process associated with the number of photons emitted by the target object.
So we have :
𝐸 𝑁𝑜𝑏𝑗
𝑝ℎ
𝑡 = 𝑉 𝑁𝑜𝑏𝑗
𝑝ℎ
𝑡 = 𝜆𝑜𝑏𝑗
𝑝ℎ
. 𝑡
▪ Note: expectation and variance are identical and they are linear in 𝒕.
▪ The number of electrons generated by the sensor is not exactly equal to the number of photons reaching it.
The average numbers of electrons and photons differ by a factor called Quantum Efficiency, noted 𝑄𝐸.
More specifically we have:
𝑄𝐸 =
𝐸 𝑆𝑜𝑏𝑗
𝑒−
𝑡
𝐸 𝑁𝑜𝑏𝑗
𝑝ℎ
𝑡
▪ Another way to formalize it is to assume that when a photon hits the photosite it generates an electron
with probability 𝑸𝑬 and produces nothing with probability 𝟏−𝑸𝑬. This can be modeled as follows:
𝑆𝑜𝑏𝑗
𝑒−
𝑡 = ෍
𝑘=1
𝑁𝑜𝑏𝑗
𝑝ℎ
𝑡
𝑋𝑘
45
Where 𝑁𝑜𝑏𝑗
𝑝ℎ
𝑡 follows a Poisson
process and 𝑋𝑘 a Bernoulli distribution
with parameter 𝑄𝐸.

46. Equality expectation / variance of the photoelectrons number (2/2)
▪ Under such a model (called compound Poisson), we can therefore calculate the expectation and variance
of the random variable 𝑆𝑜𝑏𝑗
𝑒−
𝑡 :
𝐸 𝑆𝑜𝑏𝑗
𝑒−
𝑡 = 𝐸 𝐸 𝑆𝑜𝑏𝑗
𝑒−
𝑡 𝑁𝑜𝑏𝑗
𝑝ℎ
𝑡 = 𝐸 𝑄𝐸 × 𝑁𝑜𝑏𝑗
𝑝ℎ
𝑡 = 𝑸𝑬 × 𝝀𝒐𝒃𝒋
𝒑𝒉
. 𝒕
And,
𝑉 𝑆𝑜𝑏𝑗
𝑒−
𝑡 = 𝐸 𝑉 𝑆𝑜𝑏𝑗
𝑒−
𝑡 𝑁𝑜𝑏𝑗
𝑝ℎ
𝑡 + 𝑉 𝐸 𝑆𝑜𝑏𝑗
𝑒−
𝑡 𝑁𝑜𝑏𝑗
𝑝ℎ
𝑡
= 𝐸 𝑁𝑜𝑏𝑗
𝑝ℎ
𝑡 × 𝑄𝐸 × (1 − 𝑄𝐸) + 𝑉 𝑄𝐸 × 𝑁𝑜𝑏𝑗
𝑝ℎ
𝑡
= 𝑄𝐸. 1 − 𝑄𝐸 . 𝜆𝑜𝑏𝑗
𝑝ℎ
. 𝑡 + 𝑄𝐸2
. 𝜆𝑜𝑏𝑗
𝑝ℎ
. 𝑡
= 𝑄𝐸. 𝜆𝑜𝑏𝑗
𝑝ℎ
. 𝑡 × 1 − 𝑄𝐸 + 𝑄𝐸
= 𝑸𝑬. 𝝀𝒐𝒃𝒋
𝒑𝒉
. 𝒕
= 𝐸 𝑆𝑜𝑏𝑗
𝑒−
𝑡
46
In conclusion, even though the
electron counting process is not
Poisson, it can be shown that its
expectation is equal to its
variance and that they are linear
in 𝒕.

47. Agenda
▪ Sensor modeling
▪ Photometry
▪ Appendix
▪ Bibliography
47

48. Bibliographie (1/2)
Conference by Robin Glover (SharpCap software developer) - Practical Astronomy Show 2019
Deep Sky Astrophotography With CMOS Cameras
https://www.youtube.com/watch?v=3RH93UvP358&list=WL&index=46
Didier Walliang (2018) - Le bruit en astrophotographie
RCE presentation November 2018
https://media.afastronomie.fr/RCE/PresentationsRCE2018/Walliang-RCE2018.pdf
Thierry Legault - Astrophotographie
Editions Eyrolles
Lecture material on Computational Imaging and Sensor Theory (Stanford University)
https://isl.stanford.edu/~abbas/ee392b/lect08.pdf
Henry Joy McCracken (2017) - Institut d’Astrophysique de Paris
An introduction to photometry and photometric measurements
http://www2.iap.fr/users/hjmcc/hjmcc-photom-ohp-2017.key.pdf
48

49. Bibliographie (2/2)
Calculate Sky Background Electron Rate – Robin Glover (SharpCap software developer)
http://tools.sharpcap.co.uk/
ESO (European Southern Observatory) website detailing in the “Exposure Time Calculators” section, the methodology for
measuring photon flux from point and extended sources
https://www.eso.org/observing/etc/doc/formulabook/index.html
La photométrie pour amateur d'astronomie
Conference by Pierre Strock - Nuit Astronomique de Touraine 2017
http://strock.pi.r2.3.14159.free.fr/Ast/Art/Photometrie/Photometrie__Article__PS__2017.05.23.pdf
Fabien Besnard (2013) - Introduction à la mécanique quantique
Lecture material on Quantum Mechanics (EPF 3rd Year)
http://fabien.besnard.pagesperso-orange.fr/cours/EPF/mecaq.pdf
Nana Engo (2019) – Oscillateur harmonique quantique
Department of Physics, Faculty of Science, University of Yaounde I
https://www.researchgate.net/publication/337479704_Oscillateur_harmonique_quantique
49