Compare the imaging performance of 2 telescopes for astrophotography.
Performance indicators are: pixel scale (res), FOV, extended object irradiance (eoi), point object irradiance (poi), etendue (e), pixel etendue (pe), pixel signal (ps) and object signal (os).
Compare a 100mm aperture f/6 with an 80mm aperture f/7 :
compare-telescopes.py --d1 100 --f1 6 --d2 80 --f2 7
Telescope 1 f/6.00 f= 600mm D=100mm O= 0% res=1.31"/p FOV=22'x22'= 0.87x eoi= 1.36x poi= 2.13x et= 1.36x pet= 1.36x psi= 1.36x
Telescope 2 f/7.00 f= 560mm D= 80mm O= 0% res=1.40"/p FOV=23'x23'= 1.15x eoi= 0.73x poi= 0.47x et= 0.73x pet= 0.73x psi= 0.73x
The larger and f/6 telescope is 1.36x faster than the smaller and f/7 one (but this is not the whole story, look at the other examples).
No camera is specified here, one is made up for the comparison with 1000x1000 pixels, 4μm pixel size, 100% QE.
For US-folk: replace --d1 with --di1 which accepts inches.
The program comes in 2 versions; an online web version and a python command line version.
For example compare these two telescopes :
- 10" LX200, f = 2970mm, central obstruction 37%, KAF-16803 camera 4096x4096, 9μm pixels, QE 60%
- 102mm Stellarvue SV102ED, f-ratio = 6.95, ASI1600MM camera 4656x3520, 3.8µm pixels, QE 75%
compare-telescopes.py --di1 10 --l1 2970 --o1 0.37 --c1h 4096 --c1v 4096 --c1p 9 --c1q 0.60 --d2 102 --f2 6.95 --c2h 4656 --c2v 3520 --c2p 3.8 --c2q 0.75
Telescope 1 f/11.69 f=2970mm D=254mm O=37% res=0.63"/p FOV=43'x43'= 0.33x eoi= 0.35x poi= 1.89x et= 1.75x pet= 1.71x psi= 1.37x
Telescope 2 f/6.95 f= 709mm D=102mm O= 0% res=1.11"/p FOV=86'x65'= 3.06x eoi= 2.83x poi= 0.53x et= 0.57x pet= 0.58x psi= 0.73x
The LX200 has both more resolution per pixel, and 1.37x more signal thanks to the large camera pixel size than the SV102ED.
This changes completely around when we use the ASI1600MM also on the LX200 :
compare-telescopes.py --di1 10 --l1 2970 --o1 0.37 --c1h 4656 --c1v 3520 --c1p 3.8 --c1q 0.75 --d2 102 --f2 6.95
Telescope 1 f/11.69 f=2970mm D=254mm O=37% res=0.26"/p FOV=20'x15'= 0.06x eoi= 0.35x poi= 1.89x et= 0.30x pet= 0.30x psi= 0.30x
Telescope 2 f/6.95 f= 709mm D=102mm O= 0% res=1.11"/p FOV=86'x65'=17.55x eoi= 2.83x poi= 0.53x et= 3.28x pet= 3.28x psi= 3.28x
Now the LX200 has an unrealistic high resolution of 0.26"/pixel, and the refractor gets 3.28x more signal at the sensor.
Note that the second camera arguments were not given in which case those of the first camera are copied internally.
Of the f-ratio, aperture diameter and focal length only 2 can be specified at the same time, the program then calculates the third.
Many other parameters are optional.
- 14" Celestron, f/10.8, central obstruction 32%, total transmittance 0.85, KAF-16803 camera 4096x4096, 9μm pixels, QE 65%
- 11" Celestron, f/10, central obstruction 31%, total transmittance 0.85, ASI1600MM camera 4656x3520, 3.8µm pixels, QE 75%
compare-telescopes.py --di1 14 --f1 10.8 --o1 0.32 --t1 0.85 --c1h 4096 --c1v 4096 --c1p 9 --c1q 65 --di2 11 --f2 10 --o2 0.31 --t2 0.85 --c2h 4656 --c2v 3520 --c2p 3.8 --c2q 75 --detail
OTA 1 resolving power 0.354 [arcsec], plate scale 53.708 [arcsec/mm] = 18.6 [μm/arcsec]
OTA 1 focal ratio f/10.8, focal length 3840 [mm], aperture diameter 356 [mm], central obstruction ratio 0.32, diameter 114 [mm]
OTA 1 aperture area 89144.84 [mm^2], collects 1.61x more photons
Camera 1 pixel size 9.00 [μm], sensor size 4096x4096 [pixels*pixels], 36.9x36.9 [mm*mm], sensor area 1358.95 [mm^2] =5.74x larger
Camera 1 quantum efficiency factor 65.00
Telescope 1 resolution 0.48 [arcsec/pixel], FOV 33x33 [arcmin*arcmin] =3.04x larger, optical transmittance factor 0.85
Telescope 1 extended object irradiance is 0.86x more
Telescope 1 point object irradiance is 1.38x more
Telescope 1 etendue 349446.26 [m^2arcsec^2] =4.89x more
Telescope 1 pixel etendue 20828.62 [mm^2arcsec^2] =4.78x more
Telescope 1 pixel signal is 4.14x more
---
OTA 2 resolving power 0.450 [arcsec], plate scale 73.824 [arcsec/mm] = 13.5 [μm/arcsec]
OTA 2 focal ratio f/10.0, focal length 2794 [mm], aperture diameter 279 [mm], central obstruction ratio 0.31, diameter 87 [mm]
OTA 2 aperture area 55419.56 [mm^2], collects 0.62x more photons
Camera 2 pixel size 3.80 [μm], sensor size 4656x3520 [pixels*pixels], 17.7x13.4 [mm*mm], sensor area 236.66 [mm^2] =0.17x larger
Camera 2 quantum efficiency factor 75.00
Telescope 2 resolution 0.28 [arcsec/pixel], FOV 22x16 [arcmin*arcmin] =0.33x larger, optical transmittance factor 0.85
Telescope 2 extended object irradiance is 1.17x more
Telescope 2 point object irradiance is 0.73x more
Telescope 2 etendue 71479.81 [m^2arcsec^2] =0.20x more
Telescope 2 pixel etendue 4361.42 [mm^2arcsec^2] =0.21x more
Telescope 2 pixel signal is 0.24x more
The 11" focal ratio may be faster but the 14" gets 4.14x more signal at a sensor pixel.
compare-telescopes.py --help
usage: compare-telescopes.py [-h] [--just_numbers] [--brief] [--detail]
[--legend] [--formulas] [--d1 D1] [--di1 DI1]
[--o1 O1] [--l1 L1] [--f1 F1] [--r1 R1] [--t1 T1]
[--c1h C1H] [--c1v C1V] [--c1p C1P] [--c1q C1Q]
[--c1b C1B] [--d2 D2] [--di2 DI2] [--o2 O2]
[--l2 L2] [--f2 F2] [--r2 R2] [--t2 T2]
[--c2h C2H] [--c2v C2V] [--c2p C2P] [--c2q C2Q]
[--c2b C2B]
Compare the imaging performance of 2 telescopes for astrophotography.
Performance indicators are: pixel scale, FOV, extended object irradiance, point object irradiance, etendue, pixel etendue and pixel signal.
Version 1.3 add ObjectSignal as os, rename et->e pet->pe, psi->ps
Version 1.2 add defaults for aperture diameter, focal length, focal ratio
Version 1.1 pixelEtendue renamed to pet, added Etendue (of the whole system), added camera binning
Version 1.0
Source code at https://github.com/d33psky/compare-telescopes/
optional arguments:
-h, --help show this help message and exit
--just_numbers Output just the numbers
--brief Brief output
--detail Detail output
--legend Legend
--formulas Show the used formulas
--d1 D1 Telescope 1 aperture Diameter [mm]
--di1 DI1 Telescope 1 aperture Diameter [inch]
--o1 O1 Telescope 1 central Obstruction ratio [float, 0-1]
--l1 L1 Telescope 1 focal Length [mm]
--f1 F1 Telescope 1 Focal ratio, defined as focal Length / aperture
Diameter [dimensionless]
--r1 R1 Telescope 1 focal Reducer factor [float]
--t1 T1 Telescope 1 total Transmittance factor [float, 0-1]
--c1h C1H Camera 1 Horizontal pixels [count]
--c1v C1V Camera 1 Vertical pixels [count]
--c1p C1P Camera 1 Pixel size [μm]
--c1q C1Q Camera 1 QE ratio [float, 0-1]
--c1b C1B Camera 1 binning factor [integer, 1-]
--d2 D2 Telescope 2 aperture Diameter [mm]
--di2 DI2 Telescope 2 aperture Diameter [inch]
--o2 O2 Telescope 2 central obstruction ratio [float, 0-1]
--l2 L2 Telescope 2 focal Length [mm]
--f2 F2 Telescope 2 Focal ratio, defined as focal Length / aperture
Diameter [dimensionless]
--r2 R2 Telescope 2 focal Reducer factor [float]
--t2 T2 Telescope 2 total Transmittance factor [float, 0-1]
--c2h C2H Camera 2 Horizontal pixels [count]
--c2v C2V Camera 2 Vertical pixels [count]
--c2p C2P Camera 2 Pixel size [μm]
--c2q C2Q Camera 2 QE ratio [float, 0-1]
--c2b C2B Camera 2 binning factor [integer, 1-]
Use --formulas to read about the math behind the performance indicators.
Pixel Scale, or pixel resolution, is the solid angle that is projected on a single pixel.
It is measured in arcseconds per pixel, ["/pixel].
Formula:
pixel scale ["/pixel] = 206.265 [k"] * pixel size [μm/pixel] / focal length [mm]
With 206.265 the amount of arcseconds per radian / 1000 .
And arcseconds per radian = (360 / (2 * pi)) * 60 * 60 = 206264.80624709635515795...
FOV, Field Of View, is the solid angle that is projected on the camera sensor.
Formula:
angle_x ["] = camera_pixels_x [pixels] * pixel scale ["/pixel]
angle_y ["] = camera_pixels_y [pixels] * pixel scale ["/pixel]
FOV is displayed in arcminutes [']=["/60].
Extended Object Irradiance is the radiant flux (power) received by the sensor per unit area of an extended object.
Extended Object Irradiance is measured in [W/m^2].
We do not compute the irradiance itself because the ratio suffices and that varies as the inverse square of the focal ratio.
Aperture size alone does not matter for Extended Object Irradiance, only focal ratio does. (Aperture size does matter for Point Object Irradiance).
An extended object is anything that is not a point source, where a point source can be a star or anything else close to the size of the angular PSF projected onto the sky.
Formula:
Extended_Object_Irradiance_ratio = 1 / (focal ratio of ota 1/focal ratio of ota 2)^2
The Extended Object Irradiance is also known as the Speed of a film camera where an f/4 is twice as fast as an f/5.6, meaning you need only half the time.
Point Object Irradiance is the radiant flux (power) received by the sensor per unit area of a point object. For point objects such as stars the image irradiance varies as the aperture area ratio and the inverse square of the focal ratio. Aperture size matters for Point Object Irradiance, as well as focal ratio. (Aperture size alone does not matter for Extended Object Irradiance).
Formula:
Point_Object_Irradiance_ratio = (ota 1 aperture area/ota 2 aperture area) * 1 / (focal ratio of ota 1/focal ratio of ota 2)^2
Etendue is a measure of the flux gathering capability of the optical system onto the sensor. It is a purely geometric quantity.
Formula:
etendue = aperture_area [m^2] * FOV ["^2]
Pixel Etendue is the etendue for a single pixel. It represents the light-gathering power of a single pixel.
Formula:
pixel_etendue = aperture_area [mm^2] * pixel_scale^2 ["^2]
Pixel Signal is the Pixel Etendue corrected for the sensor Quantum Efficiency and total optical system Transmittance losses.
Formula:
pixel_signal = pixel_etendue * QE-factor * Transmittance-factor
Object Signal is based on the Etendue of an extended object that fits in the FOV of both scopes, corrected for the sensor Quantum Efficiency and total optical system Transmittance losses.
object_signal = aperture_area [m^2] * QE-factor * Transmittance_factor