How to Compare CameraA+ScopeX with CameraB+ScopeY

[simon hicks]

Hi all,

I have been trying to come up with mathematical relations to compare the performance of various camera/scope combinations. Essentially asking the question “For a given field of view is it better to have camera A combined with scope X or is it better to have camera B combined with scope Y?”.

The criteria for me is the field of view as the starting point, i.e. I have a given field of view that I want. I am therefore trying to compare different camera/scope combinations that achieve that given field of view and see which camera/scope combination gives me the best performance. Here I am thinking of performance in terms of getting the shortest optimum exposure time to achieve a given S/N for faint nebulae. This analysis doesn’t consider which camera is the best, or which scope optics are best, flatness of image etc, it is purely related to getting the shortest optimum exposure time for a given field of view.

I started with Steve Cannistra’s web pages and his equations relating focal length, aperture and optimum exposure time. ( http://www.starrywonders.com/fratio.html )

Basically, they show that

            1. ExposureTime is proportional to F#^2 / Aperture^2

This basically describes the way the optics work, i.e. the more aperture you have then the more signal you collect, and the lower the f# the more you compress that light into a smaller image circle, i.e. you increase the density of light in a given area. Obviously, for a reflector Aperture^2 becomes PrimaryDiameter^2 – SecondaryDiameter^2, but lets keep it simple and stick with refractors for the time being.

So the above allows us to compare say an f5 8 inch scope to an f4 6 inch scope. But the comparison is only valid for the same sensor, it basically says nothing about the camera. So to include the sensor in the comparison I have to include more terms.

The first thing we can add into the mix is the pixel size. The above assumes we are creating an image circle collected from an area of the sky and thrown onto an area at the image plane, some of that area is covered by the sensor, and a small percentage of the sensor is made up of each pixel. If the pixel is huge then it gathers a larger percentage of the image circle, and if it is smaller then a smaller region. So it seems that the relationship for exposure time for a single pixel would be;

            2. ExposureTime is proportional to 1/PixelArea

And as most pixels are square, we can write;

            3. ExposureTime is proportional to 1/PixelSize^2

We can now consider the sensitivity of each pixel, or the QE. The bigger the QE the more signal we get from each unit of light falling on the pixel resulting in a smaller exposure time. So here;

            4. ExposureTime is proportional to 1/QE

The other thing that Steve Cannistra found here http://www.starrywonders.com/snr.html is that for a given camera and optics, and for the imaging of faint nebula, and for systems where the dark noise is insignificant (cooled), then the optimum subexposure time to reach a given S/N is determined by the read noise (RN) of the camera by the relation;

            5. SubExposureTime is proportional to RN^2

[Note that Chuck Anstey has done a more detailed analysis of optimum sub-exposure time that includes the signal level of the faint nebula features that you are trying to capture here…http://www.cloudynights.com/page/articles/cat/articles/astrophotography/finding-the-optimal-sub-frame-exposure-r1571 . However, in this analysis I am assuming that when comparing CameraA+ScopeX with CameraB+ScopeY then the target is the same, and therefore becomes irrelevant in the comparison]

Now this is where I try to combine everything together. I am basically assuming that the optimum sub exposure time of relation 5 will also be determined by the other relationships 1, 3 and 4. So this suggests that;

6. SubExposureTime is proportional to (F#^2 x RN^2) / (Aperture^2 x QE x PixelSize^2)

This would seem to allow me to compare CameraA+OpticsX with CameraB+OpticsY, at least to some level where we are concerned soley about the optimum exposure time required to capture a single sub of a given FOV of a faint nebula and getting a given level of S/N.

Is the above valid? Are there more terms that should be in there? I’m really not sure, so I would welcome some input. The dimensions don’t quite seem right for a start.

If I assume that the above is valid, then as an example of its use I could compare a couple of different camera/scope combinations. Let’s assume that my starting point is that I want a 5.5 degree FOV. I have picked a couple of camera scope combinations that allow me to achieve that….other combinations are out there and will vary in terms of price, weight, size, quality, etc….these are just examples.

Lets start with the QSI1620 camera combined with a Rokinon 135mm lens, a combination that gives a FOV of 5.59 degrees along it long side. The list below shows the numbers for this combination;

Camera:                                    QSI1620

Pixel Number X                         4250

Pixel Number Y                         2838

Pixel Count (MP)                       12.1

Pixel Size (um)                          3.1

Sensor Size X (mm)                   13.18

Sensor Size Y (mm)                   8.80

QE (0 to 1)                                0.77

Read Noise (e-)                         3.5

 

Scope                                      Rokinon 135mm F2

Focal Length (mm)                    135

Operating Focal Ratio                2.8

Aperture (mm)                           48.2

 

FOV X (degrees)                       5.59

FOV Y (degrees)                       3.73

Pixel Res (arc sec per pixel)       4.74

 

Optimum SubExposureTime      0.00559

 

Note in the above that the optimum sub exposure time (calculated from relation 6) is not in units of seconds or minutes, its just a number that we can compare with another camera/scope combination.

So the second camera scope combination is the Moravian G3-16200 combined with a Borg BO7139, which has a focal length of 280mm. Note the longer focal length is needed because the Moravian camera has a larger sensor. So the numbers here are;

Camera:                                    Moravian G3-16200

Pixel Number X                         4540

Pixel Number Y                         3640

Pixel Count (MP)                       16.5

Pixel Size (um)                          6.00

Sensor Size X (mm)                   27.24

Sensor Size Y (mm)                   21.84

QE (0 to 1)                                0.56

Read Noise (e-)                         10

 

Scope                                      Borg BO7139

Focal Length (mm)                    280

Operating Focal Ratio                3.9

Aperture (mm)                           71

 

FOV X (degrees)                       5.57

FOV Y (degrees)                       4.47

Pixel Res (arc sec per pixel)       4.42

 

Optimum SubExposureTime      0.01497

 

Note that both camera scope combinations give roughly the same FOV and roughly the same pixel resolution (that’s why they were chosen). However, the relationships suggest that the Moravian/Borg combination would require sub exposure times of around 2.7 times longer than the QSI1620/Rokinon combination.

Is this a valid analysis? Or have I gone wrong somewhere in the logic or missed out some important terms? Has this all been done before more rigorously? (Please point me in the right direction!)

It would be good to have a way to compare these things....and I'm beyond the limit of my ken. :-)

Cheers

            Simon

[Dr_Ju_ju]

have a look at this site http://www.12dstring.me.uk/fovcalc.php, just dial in your equipment & get an idea of what you'll get....

[hughgilhespie]

Hi Simon,

An interesting analysis. My concern would be that as far as I can see, it doesn't take account of the size of the fully illuminated area from the scope. Presumably a 'balanced' system should have a reasonable match between size of fully illuminated area and sensor size?

Apologies if your formula does account for this - I am also way out of my depth!

Regards, Hugh

[simon hicks]

1 hour ago, Dr_Ju_ju said:

have a look at this site http://www.12dstring.me.uk/fovcalc.php, just dial in your equipment & get an idea of what you'll get....

Hi Julian,

Yup, thanks for the link. familiar with this site. Unfortunately it just calculates FOV and pixel resolution etc. It doesn't say anything about which combination would be better than another. For instance I could have two different cameras with greatly differing pixel and sensor sizes, and I would couple each of these to two completely different focal length scopes to give exactly the same FOV and pixel resolution. But which one performs better.....that's the point of the analysis.

Cheers

          Simon

[simon hicks]

49 minutes ago, hughgilhespie said:

Hi Simon,

An interesting analysis. My concern would be that as far as I can see, it doesn't take account of the size of the fully illuminated area from the scope. Presumably a 'balanced' system should have a reasonable match between size of fully illuminated area and sensor size?

Apologies if your formula does account for this - I am also way out of my depth!

Regards, Hugh

Hi Hugh,

Thanks for the reply. Yes, you are correct, the analysis doesn't take the fully illuminated image circle into account. That's one of the (many) things that are in that 'catch-all' of quality, flatness, price, etc, i.e. things you would need to consider separately. However, I would point out that both of the lens/scopes in the example above are rated at 'full-frame' so both should be fine with either of the sensors in the two cameras. This does raise the interesting question of whether (all other things being equal) if the smaller sensor would be better....because its only using the central area of an image circle and therefore should inherently give a 'flatter' image than say a full-frame sensor that is putting a lot more demands on the off-axis performance of the lens. Remember that both combinations are giving the same image FOV and pixel resolution....so the old adage of 'bigger sensor=bigger FOV' isn't applicable here.

Cheers

          Simon

[ollypenrice]

What about well depth? Bigger pixels tend to have deeper wells allowing for the capture of a higher dynamic range. Shallower wells impose shorter exposures to avoid saturation and so negate some of the advantages in QE, I'd have thought.

The problem with any theoretical analysis like this is that its conclusions will probably be invalidated by a host of practical side effects. (I appreciate that this might not be your concern in this case but I'll follow the idea briefly.) If you were to consider imaging at about 500mm FL and compared a Hyperstar with a Tak FSQ106 the Hyperstar would easily win on the numbers. It should have more resolution and be at least 6 times faster. In reality it may indeed be 6x faster but it has little hope of approaching the Tak on resolution.

Olly

[simon hicks]

5 hours ago, ollypenrice said:

What about well depth? Bigger pixels tend to have deeper wells allowing for the capture of a higher dynamic range.

Hi Olly,

You are right that dynamic range is much more important than well depth. And dynamic range is -20log10(WellDepth/ReadNoise). What you find is that the newer smaller Sony type pixels are giving much smaller read noise than the larger older pixel sensors, and in fact the dynamic range can be just as high, and often higher, with the smaller pixel sensors. For example, the Moravian 16200 with its 6um pixels has a (relatively) massive well depth of 41,000 and a read noise of 10e-, giving a dynamic range of 72.26dB, whereas the ZWO ASI 1600 with its puny 3.8um pixels has a well depth of 20,000, but a read noise of 3.5e- giving a dynamic range of 75.14dB. Conclusion is that the latter has a bigger dynamic range than the former even though the pixels are smaller. The trend seems to be that the smaller pixel sensors are coming with lower read noise.

5 hours ago, ollypenrice said:

Shallower wells impose shorter exposures to avoid saturation and so negate some of the advantages in QE, I'd have thought.

Well yes, but this is only relevant if you only take one exposure length subs to capture the whole scene (in other words you're happy with saturated stars). If you take different length exposures and combine in an HDR image then the shorter time due to the higher QE is surely a benefit. Your longest subs will be the ones to try and capture the faintest details in the nebulosity, and this length will not depend on well depth, only QE and read noise (and sky background and the level of the target of course). 

And yes, I agree with your comments about the merits of a Hyperstar vs a Tak FSQ106.....the analysis above is really just about speed and can't really say anything about the quality of the optics. The point is that it is easy to compare the speed of say an F8 200mm aperture scope with an F5 130mm aperture scope for instance....but what about the speed comparison when you place two different cameras (different pixel sizes, QEs, ReadNoise, etc) on these scopes....what will the comparison on speed be then?

A more interesting comparison would be something like a Tak FSQ106 on a 'larger pixel' camera, in order to give a FOV of say 2 degrees, vs a 'smaller pixel' camera with a shorter focal length Borg lens giving exactly the same FOV and exactly the same arc seconds per pixel. If we assume that the Tak and the Borg are of a similar quality in terms of their optics, then which combination is better (in terms of speed).....that's the point of the analysis.

Cheers

          Simon

[simon hicks]

Hi Olly,

OK, you've got my interest up now! Lets use relationship 6 shown above to compare a Tak FSQ106+camera combination with a Borg+camera combination.

Lets assume we are taking the Tak with the reducer, so its F3.6 and FL=381.6mm. And lets combine it with the Moravian G3-16200 which is a nice big 16.5MP camera. This combination gives us the following numbers;

            Camera:                                    Moravian G3-16200

Pixel Number X                         4540

Pixel Number Y                         3640

Pixel Count (MP)                       16.5

Pixel Size (um)                          6.0

Sensor Size X (mm)                   27.24

Sensor Size Y (mm)                   21.84

QE (0 to 1)                                0.56

Read Noise (e-)                         10

 

Scope                                      Tak FSQ106 plus Reducer

Focal Length (mm)                    381.6

Operating Focal Ratio                3.6

Aperture (mm)                           106

 

FOV X (degrees)                       4.09

FOV Y (degrees)                       3.28

Pixel Res (arc sec per pixel)       3.24

 

Optimum SubExposureTime      0.00572

 

We will now pick a camera with smaller pixels, which will mean a smaller focal length Borg scope to achieve the same field of view. So we'll use the ZWO ASI 1600 again and combine it with a Borg BO6738 (F3.8, FL=255mm), and the numbers are;

 

Camera:                               ZWO ASI 1600MM

Pixel Number X                         4656

Pixel Number Y                         3520

Pixel Count (MP)                       16.4

Pixel Size (um)                          3.8

Sensor Size X (mm)                   17.69

Sensor Size Y (mm)                   13.38

QE (0 to 1)                                0.77

Read Noise (e-)                         3.5

 

Scope                                      Borg BO6738

Focal Length (mm)                    255

Operating Focal Ratio                3.8

Aperture (mm)                           67

 

FOV X (degrees)                       3.98

FOV Y (degrees)                       3.01

Pixel Res (arc sec per pixel)       3.07

 

Optimum SubExposureTime      0.00354

 

Remember that both of the examples above give exactly the same FOV (within a gnat's whisker) and almost exactly the same pixel resolution (both are 16MP cameras), but the Borg/ZWO combination is 1.6 times faster than the Tak/Moravian combination.

 

There are obviously other considerations here...like the Tak+Reducer+Moravian is about three times the price and 3 times the weight of the Borg/ZWO combination. And the Tak optics might out perform the Borg....I really don't know. But the point is that the above gives a way of considering one aspect of the two combinations, i.e. the relative optimum sub exposure time.

 

That is of course if the relationships are correct! :-) And that's what I'm trying to understand.

 

Cheers

          Simon

 

[ollypenrice]

I can't comment on the theory but the obvious difference between the Tak and the Borg is that the Tak will focus far more of the blue spectrum than the Borg.

Again, not what you are really interested in here but there has been considerebale experimentation and discussion over the poor performance of the TEC140's correction with Sony sensors as compared with its excellent correction with Kodaks. So many things are out to get us! How many other refractors might vary with choice of chip I don't know, but I doubt this will really be unique to the TEC.

Olly