Ye old sampling rate issue

[Vox45]

So, thanks to the forum's posters and especially to JamesF detailled explanations and tips, I was able to take my first planetary image

I did so by just inserting my cam in the diagonal as it looked a bit bigger (directly in the visual back Saturn seemed too small on the sensor). I did not use a barlow yet, I am still waiting to get the 3x barlow I ordered.

Alaways wanting to better myself, I started to look at the sampling rate I should image at. Of course there are a thousand things more important (FL, seeing, polar alignement, filters, etc) that could better that image but I have a vague enough understanding of all those.

But sampling rate ? not so much :/

(1) I started my journey by calculating the resolution power of my scope using this formula:

[Dawes limit devided by the Diameter]

PR= 120/D

PR = 120/150 = 0.8 arcsec

I know that we should sample at twice this result (Nyquist's Sampling Theorem) so I came up with 1.6 arcsec/pixel

(2) Once I found the 'sampling rate' I need to know at wich FL I'll achieve the correct size of the object on the sensor.

For a 1500mm FL scope like mine and a 5.6micron pixel size for my webcam, I used this formula:

Resolution = (CCD Pixel Size / Telescope Focal Length ) * 206.265

arcsec/pixel = (5.6/1500)* 206.265 = 0.77 arcsec/pixel ... that is not even near my resolution power of 1.6 !

I tried a FL of 1000

arcsec/pixel = (5.6/1000)* 206.265 = 1.6 arcsec/pixel ???? whoa what ? I need to REDUCE my focal ?

but what about the "rule of thumb that can be used as a starting point which is that the focal ratio of the entire optical train needs to be five to six times the numerical value of the pixel size in thousandths of a millimetre" ... that would mean that I need to use a F/D of 5.6*5 = 28 which translate to a FL of : 28*150 = 4200 !

Putting that 4200 FL into the sampling rate I get:

arcsec/pixel = (5.6/4200)* 206.265 = 0.27 arcsec/pixel

So now I am very very confused !

I may be way off in my calculation and/or mixing concepts and formulas ... If anyone was brave enough to read all this and have an idea where I went wrong, I would be glad to read from you !

People may say that I'm splitting hairs; quite true but I don't know any other fun way of learning

P.-S. I did use this website to verify some of the math there. Quite a useful tool

[Cornelius Varley]

"I know that we should sample at twice this result (Nyquist's Sampling Theorem) so I came up with 1.6 arcsec/pixel"

You should divide by 2 not multiply to get the sampling rate.

The last equation is correct. For lunar/planetary imaging you need to use somewhere between 0.38 and 0.27 arcsec/pixel with that combination of camera and telescope.

[JamesF]

Where you correct for Nyquist in 1), you appear to have halved the sample rate, not doubled it.  If your "base" scale is 0.8 arcseconds/pixel then to double the sampling rate you need to spread that 0.8 arcseconds over two pixels, so your scale becomes 0.4 arcseconds per pixel which gets you into the same kind of area as 2).

I'm not confident of using Dawes Limit for the basis of the calculation personally.  Rayleigh's formula says that resolution also depends on wavelength and the wavelength of the visible spectrum varies by around a factor of two from one end to the other.  It's possible that the Dawes Limit is only correct for the mid-point of the spectrum or something like that.  In fact, since it should be easy I've just roughly guessimated that the Dawes Limit agrees with Rayleigh at around 560nm which is towards the blue end of the "green" part of the spectrum.

James

[pete_l]

I know that we should sample at twice this result (Nyquist's Sampling Theorem) so I came up with 1.6 arcsec/pixel

Nyquist's theorem defines the lowest sampling rate needed to recover a signal - and even then it relies on low-pass filtering to remove higher-order effects. There is no upper limit. So "according to Nyquist" :grin:   you could image at any pixel scale smaller than your telescope / viewing / camera resolution and you'll be fine. Since you'll be stacking the images, you can probably image at pixel scales above your telescope#s resolution, too.

[Vox45]

"I know that we should sample at twice this result (Nyquist's Sampling Theorem) so I came up with 1.6 arcsec/pixel"

You should divide by 2 not multiply to get the sampling rate.

The last equation is correct. For lunar/planetary imaging you need to use somewhere between 0.38 and 0.27 arcsec/pixel with that combination of camera and telescope.

Ho silly me ! Am I red in the face now

Ho well we learn something new everyday

Where you correct for Nyquist in 1), you appear to have halved the sample rate, not doubled it.  If your "base" scale is 0.8 arcseconds/pixel then to double the sampling rate you need to spread that 0.8 arcseconds over two pixels, so your scale becomes 0.4 arcseconds per pixel which gets you into the same kind of area as 2).

I'm not confident of using Dawes Limit for the basis of the calculation personally.  Rayleigh's formula says that resolution also depends on wavelength and the wavelength of the visible spectrum varies by around a factor of two from one end to the other.  It's possible that the Dawes Limit is only correct for the mid-point of the spectrum or something like that.  In fact, since it should be easy I've just roughly guessimated that the Dawes Limit agrees with Rayleigh at around 560nm which is towards the blue end of the "green" part of the spectrum.

James

I can alway count on a clear explanation from you JamesF

Nyquist's theorem defines the lowest sampling rate needed to recover a signal - and even then it relies on low-pass filtering to remove higher-order effects. There is no upper limit. So "according to Nyquist" :grin:   you could image at any pixel scale smaller than your telescope / viewing / camera resolution and you'll be fine. Since you'll be stacking the images, you can probably image at pixel scales above your telescope#s resolution, too.

Again, splitting hairs is my specialty but at least from now on I'll have a better understanding... Just need to clarify what is the actual effect of bining !