Focal Ratio equation

[nmoushon]

I was wondering what the actually equation is to figure out the difference in imaging time that is needed if I where to image at a slow f/ratio vs a faster f/ratio. For example how much for time would I need to image at F/10 to equal imaging at F/8.

Also is there a seperat equation for comparing single subs vs total exposure time or is it the same equation?

[JamesF]

The difference will be the ratio of the squares of the f-number for the same level of exposure, so f/10 will need ( 10 x 10 ) / ( 8 x 8 ), or about 1.6 times as long as f/8.

Not sure I understand the second question though.

James

[thanders]

(10/8) to the power of 2 = 1,56 times longer exposure with f10

/Thomas

[nmoushon]

Thank James. Sorry if the second question was confusing. I'll try again.

So the equation shows that at F/10 you need 1.6x to equal F/8. From what I previously understood that means that an indivual sub at F/10 needs to be 1.6x longer that a sub at F/8 to aquire the same amount of data. Now does that apply to total exposure time as well? So if I did a total of 4hrs at F/8 do I use the same 1.6x to find my total hr at F/10?

[JamesF]

By way of explanation, this is because the focal ratio deals with linear dimensions whereas exposure depends on the area over which the photons are falling.  If you increase the linear dimensions by factor of 10/8 (by moving from f/8 to f/10) then the area on which the photons are falling will increase by a factor of 10/8 on both axes, so you need 10/8 more time to account for the increase in the "horizontal" axis, and 10/8 more again to account for the "vertical" axis as well.

James

[JamesF]

Thank James. Sorry if the second question was confusing. I'll try again.

So the equation shows that at F/10 you need 1.6x to equal F/8. From what I previously understood that means that an indivual sub at F/10 needs to be 1.6x longer that a sub at F/8 to aquire the same amount of data. Now does that apply to total exposure time as well? So if I did a total of 4hrs at F/8 do I use the same 1.6x to find my total hr at F/10?

I *think* that may depend on how you're combining the subs.  Each sub needs to be 1.6 times as long to achieve the same exposure, but fewer exposures means read-out noise may have less effect.  I'm not sure about that one.

James

[dph1nm]

Changing the f-ratio will not change the amount of light you collect from an object (providing it is entirely within the field of view of both telescopes. Only changing the aperture changes this. What it does change is the pixel scale (and hence the number of photons per pixel).

NigelM

[nmoushon]

Changing the f-ratio will not change the amount of light you collect from an object (providing it is entirely within the field of view of both telescopes. Only changing the aperture changes this. What it does change is the pixel scale (and hence the number of photons per pixel).

NigelM

I don't think this is entirely correct. A 4" f/5 and a 8" f/5 scope will collect the same amount of data. The scale of the object will be different though. I know pixel size does play into it as well, though Im not as knowledgeable on that end of things. Aperture increase does change how much light is gathered and is noticeable to our eyes as such but for electronics it doesnt work the same.

[ollypenrice]

I don't think this is entirely correct. A 4" f/5 and a 8" f/5 scope will collect the same amount of data. The scale of the object will be different though. I know pixel size does play into it as well, though Im not as knowledgeable on that end of things. Aperture increase does change how much light is gathered and is noticeable to our eyes as such but for electronics it doesnt work the same.

In my view Nigel is correct. It is simply not true that a fast F ratio necessarily reduces exposure times. This is the F ratio myth. 

A fast scope of a given aperture catches the same amount of light as a slow one. Light grasp depends only on the surface area of the clear aperture. F ratio has nothing whatever to do with this. How can it? Incident light going down the tube cannot know the F ratio in which it is about to participate.

I think the following is the only sane way to think about it;

Take a scope of 100mm aperture and 1000mm focal length, so it's F10. We want to change it to F5, so what do we do? Two choices;

Option 1)  We increase its aperture to 200mm without changing the focal length. This makes it F5. The increase in aperture makes it collect 4x as much light so it is 4x faster. It is not 4x faster because it is F5, it is 4x faster because it has a clear aperture which is 4x greater than the original scope. The proof lies in Option 2 below...

Option 2)  We put a 0.5x focal reducer in the original scope. This does not increase light grasp but what it does do is reduce focal length so that a wider area of sky lands on the chip and, yes, any light from that previously unrecorded part of the sky does land on the chip so the chip gets more light. But if your object fitted on the chip at F10 none of the new light comes from that object.  If your interest is the object itself rather than what lies around it then the new light is useless. It isn't from the object.

When you reduce focal ratio merely by reducing focal length you concentrate the scope's object photons (those from the small object) onto fewer pixels. These 'fill' faster but there are fewer of them so resolution is reduced. You can do pretty much the same thing by shooting at F10 without the reducer and then downsizing the image to a quarter the area of the original. This will have the same effect.

Conclusion; If you go from F10 to F5 by increasing aperture you get new light from the object. If you go from F10 to F5 by reducing focal length you don't get new light from the object. It really is that simple.

Olly

[IanL]

Here are a few graphs I did a while ago to illustrate matters.  The first shows how much less light would fall on an area (say a pixel in your camera) as you increase the f-ratio.  I've arbitrarily chosen f/2 as the staring point, with higher f-ratios plotted as the percentage of light falling in the same area.

The simple rule is that every time you double the f-ratio, you get a quarter of the light of the previous ratio.  So for f/4 we get 25% of the light of f/2 in the same area, and for f/8 we get 25% of the light of f/4 or 6.25% of the light of f/2 (a quarter of a quarter).

As Olly said, you can change the f-ratio by changing the aperture and keeping the focal length the same:

The same rule applies as above - half the aperture gives you a quarter of the light.  It's a simple matter of geometry - if you half the aperture of the objective, you reduce it's area to a quarter of the original:

So 200mm aperture = 31,416mm² (pi x radius²),  100mm aperture = 7,854mm².  7,854 / 31,416 = 0.25 (i.e. one quarter of the area).  A quarter of the aperture bringing light in means a quarter of the light falling on your pixel.  No magic here, move along.

If you change the f-ratio by changing the focal length and keeping the aperture constant, you get this instead:

So if you double the focal length again you get a quarter of the light in the same area.  The reason is that doubling the focal length doubles the magnification of the scope, i.e. a line that appears 1mm long at prime focus for a focal length of 500mm would appear 2mm long if you increase the focal length to 1,000mm.  Since we are considering how much light falls in an area:

- 100% of the light falls in a box of 1mm x 1mm = 1mm²

- Double the length of the sides and we have 2mm x 2mm = 4mm²

- But we have the same amount of light falling in 4 x 1mm² boxes, so a quarter of the light falls in each of the 1mm² boxes as fell in the original 1mm² box.

- Now just substitute camera pixels for boxes and you've got it!

Again it's not magic, just geometry.

With regard to combining subs, the simplistic way to think about it is just to take the total length of all the exposures.  If you take 16 x 8 minute subs (128 minutes total) then half the f-ratio, it is the equivalent  4 x 8 minute subs (32 minutes total) or 16 x 2 minute subs (again 32 minutes total).  It's not quite that simple as read noise will come in to it due to different numbers of subs, but that's the basic rule: half the f-ratio = quarter of the exposure time.  (Or if you prefer double the f-ratio = 4 x the exposure time).

Of course that doesn't mean the images will be equivalent - as previously said if you decrease the f-ratio by using a bigger aperture, you will get the same image in a quarter of the time.  If you do it by reducing the focal length, you won't - you'll end up with the same signal to noise ratio (SNR) in a quarter of the time, but the image of the target will also be smaller.

In the second case you can achieve nearly the same thing by properly down-sampling the larger image to a quarter of it's original size (i.e. halfing the height and width). Provided your downsampling is taking the average of four pixels to create one smaller one, you've put the same amount of samples in to a quarter of the area. (Again it's not precisely equivalent, but near enough as makes no difference).  Top-tip there - if you have a noisy camera (like a DSLR), process your images at full size and then display them on the web at a smaller resolution and see how much better they look!

I always think about it like a bucket left out in the rain.  If you make the bucket twice as wide, you will collect 4 x more water.  If you keep the bucket the same size but put a funnel in it that is the same size as the bucket, you can produce an impressively bigger stream of water in one spot on the bottom of the bucket, but at the end of the shower you'll still have the same amount of water in the bucket.  Focal reducers are just funnels, when what we all really want is a bigger bucket!

[ollypenrice]

Nicely put by IanL.

It's just occurred to me that normal daytime photographers adjust their F ratios by varying their clear aperture with diaphragms while telescope users adjust theirs by altering their focal lengths with extenders/reducers and that this may be the source of the F ratio myth: If, like the daytime phographers, you lower your F ratio by increasing your aperture you do indeed speed up capture, F5 being 4x faster than F10. No F ratio myth for these folks, then. What has, perhaps, happened is that telescope users have come to associate fast F ratios with short exposures when what they should be doing (when aiming to image something that will fit on the chip reduced or unreduced) is associate fast exposures with larger apertures.

The fast F ratio=shorter exposure idea comes from a camera world in which focal length is fixed and aperture is variable. The idea is incorrect or misleading if you move into a telescope world in which aperture is fixed and focal length controls F ratio.

So are focal reducers useless? If the object will fit on the chip unreduced, yes, in a nutshell, they are useless. But if you actually want the object plus the stuff around it in the final image then, no, focal reducers are great. Not only will you get a wider view but you will reach a satisfactory signal to noise ratio 4x faster at F5 than at F10.

Olly

[dph1nm]

The one thing I will say, is that I have become slightly more convinced that for typical amateur setups read-noise is a bigger issue than I once thought it was (especially with older DSLRs, which many people still have). So there may be quite significant signal-to-noise improvements to be had by having larger pixels (on the sky)  i.e. smaller f-ratio, same aperture.

NigelM

[nmoushon]

Thank you for the explanations Olly an Ian. Ian I think you meant 8x4 min not 4x8min but I understood your point.  

Everything you both have said makes sence. What I don't get is then that why do people get more data from a scope imaging at F/4 than one at F/10 for the same exposure time? Or at least thats what it definately looks like to me. For example no one I see is trying to image nebula at F/10. But from what you've just said as long as you have enough aperture it shouldnt matter. And for that case why do we not all image with as large an aperture as possible and either have a long focal length or throw on a 2x or 3x extender? (assuming everyone can afford a mount that could handle that kind of equipement)

And also is resolution really determined mainly by the aperture or is it the focal length? I'm picturing it my head as me trying to look at a painting on a wall 50 feet away and not being able to pick out much detail. So to increase how much detail I can see I now move up to 5 feet away. My eyes didnt get any bigger(relating to aperture) I just shortened my distance to the painting (relating to focal length). Or I could achieve the same thing by still standing 50 feet away and looking through a magnifing glass. Does my logic make any sence at all? lol Or does this not apply the same way with telescopes and AP?

Also I dont agree with the rain bucket analogy. If you put a funnel of 8" on a 4" bucket you will collect the same amount of water as an 8" bucket over the same amount of time. Now in reality theres different factors that could come into play such as how hard the rain is coming down and if the funnel can funnel the water through fast enough that it wont over flow but if we are to compare it to light gathering then those don't matter.

[ollypenrice]

You say, 'What I don't get is then that why do people get more data from a scope imaging at F/4 than one at F/10 for the same exposure time?' 

They get more data because their field of view is wider so more objects contribute to the light falling on the chip. I took M42 only, once, in my FSQ85 without reducer at F5.3. Then I took M42 plus the Running Man in the same scope with reducer at F3.9. This gave me more signal because I got light from the Running Man as well as M42 and M42 was pouring light onto fewer pixels which 'filled' faster. But, but, I did not get any more light from M42.

Or you can say that they get more signal because they pour more arseconds of skylight onto fewer pixels.

Also uncooled cameras like DSLRs cannot take long enough exposures at F10 to get enough signal. This is why they only compete with CCDs in ultra fast systems like the Tak Epsilon etc. But if you take long enough subs over a long enough time F10 can produce a perfectly good image. But it would be better to add aperture at the same focal length and get more signal in the time. It just gets prohibitive. Who can afford a 2.5 metre focal length at F4?

Olly

[nmoushon]

So the f/ratio, just the number, doesnt actually coorilate to how much data I'm collected for any given length in a sub? The aperture is what determines the amount of data?

So an C11 with hyperstar is going to collect WAY more data than the Tak Epsilon 130 even though there focal lengths and F/# are not that far off from each other?

I also have always understood that with enough time given you make any image taken at f/10 equal an image taken at any faster f/#. BUT it could take weeks worth of nights imaging versus a night or two. So is that not technically collecting more data faster?
 

[Alien 13]

The scope F ratio is still useful provided you keep to basics, the focal length for a given sensor size determins the image scale the F ratio of a particular scope of said focal length will then determine its speed so a 500mm FL F8 scope will be slower than a 500mm FL F4 one. The confusion comes when you mix differing sensor size and FL with with a set target image scale.

Alan

[nmoushon]

The scope F ratio is still useful provided you keep to basics, the focal length for a given sensor size determins the image scale the F ratio of a particular scope of said focal length will then determine its speed so a 500mm FL F8 scope will be slower than a 500mm FL F4 one. The confusion comes when you mix differing sensor size and FL with with a set target image scale.

Alan

See thats what I thought. But Olly an Ian are saying that the aperture is the constant not the FL. Which is why he think the f/ratio myth is around. (read a couple posts back)  Plus I dont think you can have a set image scale but differ the FL can you? I thought the sensor size was just cropping the imaging circle not changing the scale of it.

[ollypenrice]

Stick to one focal length when comparing x scope with y scope. If you do, the key thing is aperture. This can be expressed as F ratio but the key thing is aperture.

If you change aperture and focal length in one comparison you are comparing apples with oranges. Don't do it.

Yes, on paper the Hyperstar is the best thing since sliced bread but I don't want one. I might just consider an Epsilon but with any fast system you need to get it to work. Hardly anybody gets Hyperstars to do anything more than take mediocre images quickly. Expert tweakers do better. I've seen an OS Riccardi Honders close up as well and if you can get it to perform to the book, great. Neither its owner nor I managed in the time we had.

A Tak FSQ 85 with reducer at F3.9 just works.

Super fast optics are great on paper but be careful. Think about depth of field and orthogonality and collimation and you'll see what the problem is.

Olly

[Alien 13]

See thats what I thought. But Olly an Ian are saying that the aperture is the constant not the FL. Which is why he think the f/ratio myth is around. (read a couple posts back)  Plus I dont think you can have a set image scale but differ the FL can you? I thought the sensor size was just cropping the imaging circle not changing the scale of it.

The analogy with camera lenses is still valid i think at a set focal length, a 200mm F2.8 will give you exactly the same image as a 200mm F4 but will be brighter i.e.more photons due to the larger objective.

The image scale is often limited by the scopes ability to produce a flat field but if everything was perfect you could fit all of M31 into an image with a 2M focal length scope provided you had a sensor big enough.

P.S.  Olly is correct the real confusion is when you try and compare" apples and oranges" for example my scope has a faster F ratio than say an ED80 but it has a smaller diameter objective so with both taking an image of M42 with the same sensor who gets the most photons in a given time?

Alan

[IanL]

Thank you for the explanations Olly an Ian. Ian I think you meant 8x4 min not 4x8min but I understood your point.  

Everything you both have said makes sence. What I don't get is then that why do people get more data from a scope imaging at F/4 than one at F/10 for the same exposure time? Or at least thats what it definately looks like to me. For example no one I see is trying to image nebula at F/10. But from what you've just said as long as you have enough aperture it shouldnt matter. And for that case why do we not all image with as large an aperture as possible and either have a long focal length or throw on a 2x or 3x extender? (assuming everyone can afford a mount that could handle that kind of equipement)

And also is resolution really determined mainly by the aperture or is it the focal length? I'm picturing it my head as me trying to look at a painting on a wall 50 feet away and not being able to pick out much detail. So to increase how much detail I can see I now move up to 5 feet away. My eyes didnt get any bigger(relating to aperture) I just shortened my distance to the painting (relating to focal length). Or I could achieve the same thing by still standing 50 feet away and looking through a magnifing glass. Does my logic make any sence at all? lol Or does this not apply the same way with telescopes and AP?

Also I dont agree with the rain bucket analogy. If you put a funnel of 8" on a 4" bucket you will collect the same amount of water as an 8" bucket over the same amount of time. Now in reality theres different factors that could come into play such as how hard the rain is coming down and if the funnel can funnel the water through fast enough that it wont over flow but if we are to compare it to light gathering then those don't matter.

8 x 4mins = 32mins, 4 x 8mins = 32mins, 16 x 2mins = 32mins 1 x 32mins = 32mins and they're all one quarter of 128mins.  In terms of photons sampled, it makes no difference - half the f-ratio = one quarter of the total exposure time.  It does matter in terms of read noise, and it also matters in terms of pixel rejection algorithms (more subs is worse for read noise but better when you are trying to reject hot pixels, plane trails and cosmic ray hits - where the balance lies is a conversation for another day I think )

The reason you don't agree with the bucket analogy is that you're misconstruing it.  If you put an 8" funnel on a 4" bucket you have got an 8" bucket not a 4" bucket, so of course you'll get more water.  I.e. you have moved from an 4" scope to an 8" one, and that is exactly the point - bigger aperture = more photons collected overall, focal reducer = same number of photons concentrated in a smaller area.  As we have said several times, if your target already fits on the chip at the unreduced focal length, you won't gain any more data by using a reducer.

You are correct that the resolving power of a scope is ultimately governed by aperture - bigger aperture = more resolving power.  But that isn't really the issue when imaging, what matters is how many arcseconds fit in to a single pixel, i.e. the image scale.  Image scale is governed by focal length and the size of the pixels.  Bigger pixels = more arcseconds per pixel (i.e. a less detailed image), shorter focal length also = more arcseconds per pixel and less detail.

In theory, once you reach a pixel scale that less than twice the resolving power of your aperture, you are oversampling the image and you won't gain any more detail in the image.  Conversely if you have a pixel scale that is more than twice the resolving power you are undersampling the image and you have less detail than you might.

In practice, your resolving power is more likely to be limited by the local seeing conditions than by the resolving power of your scope, so you'd usually work out an ideal pixel scale relative the best seeing conditions you typically get.  In the UK that is typically 2 arcseconds but your mileage will vary. So you'd want a pixel scale of 1 arcsecond per pixel to optimally sample it (read up on Nyqist if you want to know why).

Once you know the ideal pixel scale, you try to match a scope and camera to achieve it.  This is easier said than done and most of the time you should just worry about finding a cameras/scope combination with a focal length that your mount can guide reliably at, and that fits the targets you want on to the sensor with a flat fully illuminated field.  If you get that right and still have a choice of equipment left, then consider the pixel scale in terms of over/under sampling.

So the f/ratio, just the number, doesnt actually coorilate to how much data I'm collected for any given length in a sub? The aperture is what determines the amount of data?

So an C11 with hyperstar is going to collect WAY more data than the Tak Epsilon 130 even though there focal lengths and F/# are not that far off from each other?

I also have always understood that with enough time given you make any image taken at f/10 equal an image taken at any faster f/#. BUT it could take weeks worth of nights imaging versus a night or two. So is that not technically collecting more data faster?

The f-ratio will give you a good guide as to how quickly you can reach a given signal to noise ratio (SNR).  Two different setups with the same f-ratio will achieve the same SNR in the same time, but that doesn't mean you will have the same amount of detail in the image.  SNR is driven by the number of photons captured in a pixel a given amount of time, and you could have a big aperture and long focal length to get lots of incoming photons on lots of pixels (and thus more detail) or a small aperture and short focal length concentrating fewer incoming photons on fewer pixels (and less detail), but the f-ratio basically says in either case a single pixel will receive the same number of photons and thus reach the same SNR.

Regarding the C11 vs Tak, in short no.

C11 with hyperstar = 560mm / 279mm = f/2.0

Tak = 430mm / 130mm = f/3.3

Now the difference between f/2 and f/3.3 might sound like "not that far off from each other", but if you consult my graphs above, you'll see that f/3.3 gives you less than 37% of the illumination of f/2. In other words you'd have to expose for nearly three times as long with the Tak to get the same SNR as the C11.  Despite the shorter focal length of the Tak, the aperture of the C11 wins hands down, as it has more than four and a half times the light gathering capacity of the Tak.

The field of view and pixel scale are also different:

The in my example with an Atik 460EX the Tak is undersampling if you have 2 arcsecond seeing, as is the C11 but less so.  The ideal scenario would be to use a camera with smaller pixels to get to 1 arcsecond per pixel. The Tak also has a significantly wider FOV.

Regarding imaging at f/10 again yes, you can image at any f-ratio and get the same as a lower f-ratio if you expose for longer.  In theory it's purely down to double the f-ratio = 4 x the exposure, but in practice as you go to higher and higher f-ratios the read noise becomes more of a concern and you'll need far more than 4x the exposure length to reach the same SNR.

[ollypenrice]

In terms of the numbers the C11 Hyperstar beats the Tak hands down. But will a Hyperstar really beat a Tak? Usually not. Mass produced optics, an extremely demanding lack of focal depth, an inadequate focus mechanism, the difficulty of collimation at fast F ratios... All these explain the generally indifferent standard of Hyperstar images. They also explain the occasional and remarkable exceptions when some clever soul has surmounted the practical difficulties of fine tuning a Hyperstar and managed to get it to do what the numbers say it should do.

In the end AP is a practical activity...

Olly

[IanL]

In terms of the numbers the C11 Hyperstar beats the Tak hands down. But will a Hyperstar really beat a Tak? Usually not. Mass produced optics, an extremely demanding lack of focal depth, an inadequate focus mechanism, the difficulty of collimation at fast F ratios... All these explain the generally indifferent standard of Hyperstar images. They also explain the occasional and remarkable exceptions when some clever soul has surmounted the practical difficulties of fine tuning a Hyperstar and managed to get it to do what the numbers say it should do.

In the end AP is a practical activity...

Olly

I don't think there is any dispute about the practical issues, but we were discussing the theory and I'm just going with the example provided.

[ollypenrice]

I don't think there is any dispute about the practical issues, but we were discussing the theory and I'm just going with the example provided.

Indeed, and the scopes make an interesting comparison both in terms of theory and practice.

Olly

[dph1nm]

Personally I think the term 'faster' should be banned! It implies something which is generally not true and causes no end of confusion. A small f-ratio is all about getting a wider area quicker. There is no guarantee it is going to get you as deep on a single object in a shorter time, or deeper in the same time - that is all down to whether read noise is a significant factor or not. If it isn't then f-ratio makes no real difference (for the same aperture).

NigelM

[nmoushon]

Thanks again everyone for battling through all of this with me. I guess I really fell hard for the f/ratio myth. 

Quick questio since it got brought up. I always thought that going deeper in an image required either TONS of exposure time and really long subs or having fast optics. Like trying to get the fain flux around galaxies. With my new knowledge and what Nigel just said Im still not quiet sure whats going to be the dominant factor to achieve that kind of imaging. Is it the aperture or exposure length and total time or camera sensitivity?