I am not an expert in this field, but as far as I understand, for a given observation time if I collect a total of N photons, then my signal ~ N and noise ~ \sqrt{N} (assuming Poisson noise), so the SNR ~ N/\sqrt{N} = \sqrt{N}. Now depending on the required SNR and number of photons I collect per unit time, I can choose my observation time \delta t. In all the ideal conditions that you have mentioned, it should not matter which one I choose, small aperture and long exposure vs. big aperture and short exposure. But the image may get distorted if I choose very small aperture due to the diffraction pattern from the edges of the telescope structure holding the secondary mirror, which will result in spikes in the image. So it will be better to choose short exposure if possible.

Hi:

For stars, only the aperture is important. As far as the star is not resolved, you only detect a point-like source, which on the detector has a shape given by the Point Spread Function (PSF). Out of the atmosphere, the shape of the star on the detector depends only on the diffraction pattern which is a function of the geometry of the telescope. On earth, the shape is a convolution between the diffraction pattern and the atmospheric seeing.

For a given SNR you must take into account:

- The collecting area of the telescope. For a circular optical telescope, it is simply the pi\*R^2 formula.

- The efficiency of the system: surface reflection, filter transmission, detector quantum efficiency.

- The power of the signal and the background (the background out of the atmosphere may be negligible).

- The noise: Photon (shot) noise due to the star and the background, detector and electronic noise.

The shortest exposure times may be preferred.

- Tracking is better for short exposures.

- Seeing does not vary during short exposures.

- Distortions due to environmental changes and telescope flexures are minimal during short exposures.

- There are less cosmic rays during short exposures.

- You can calculate a better statistics using as many data points as you can.

But:

- In practice, you spend more time taking 5 exposures of 1 min than taking one single exposure of 5 min, because of CCD readout.

- Total readout noise increases proportionally to the number of exposures.

Thus, you have to deal with all these issues, and select exposures that are long enough for a good individual SNR, and they are as numerous as possible. For bright stars long exposures are usually unfeasible due to CCD saturation. For dimm stars, I probably will choose single exposures with a SNR about 100 (error of about 0.01 mag), and take as many shots as I can.

How the exposure time is choosen for a given SNR or star brightness in the light frames?

You need an ETC, Exposure Time Calculator. This brings together all the observing parameters (as listed in Jose Antonio's answer) into an overall formula for signal to noise. Many of these parameters are fixed for a given instrumental setup; but there are also others which are time variable, such as seeing/point spread function, and background intensity, which for a given site is usually dominated by changes in moon phase.

Most observatories provide ETCs so that astronomers can plan their observations, and make informed requests for the amount of observing time necessary to achieve their science goals. If you google "Exposure Time Calculator", you'll see that every link for several pages is to an observatory-specific ETC. It's done this way because every telescope, camera, filter, grating, grism and spectrometer has its own unique sensitivity/transmission curve as a function of wavelength, and sky parameters such as extinction coefficients can also be site-specific.

It should be possible to build your own photometry ETC if you can approximately characterise the sensitivity of your system (optics, filters, photometry camera) and plug this data into a generic ETC engine. See https://anthonykgross.fr/data/cnrs/ADASS-XXIV_ETC-42.pdf for a recent example; or from my own previous work, see Article An ETC and Image Simulation Suite for any ground based Camer...

Vasil: The closely related question is about the ISO/gain. If we want to measure small deviations in brightness, we need more gain.

I caution that one should change ISO/gain only if doing so reduces the readout noise (in electron units) of the camera. Many cameras are "ISO-less", insofar as increasing ISO/gain only brightens the presented image by a constant factor, but does not improve the underlying signal to noise - and may in fact have the downside of saturating the bright end sooner, thus decreasing dynamic range.

Vasil: What will be the the change in the star brightness in the frames for the 2 variants: long exposure and small aperture, short exposure and large aperture? Will we get some kind of averaging in time, in the case of long exposures?

Consider that over the same observing time commitment, you can also get an "averaging in time" benefit by combining multiple short exposures through the large aperture. And this approach will yield far higher net SNR due to the total flux collected with the larger aperture.

But if the question is - "per frame, does long exposure and small aperture yield the same SNR as short exposure and large aperture?" - the answer is also "no". We can consider two scenarios:

Vasil: "1) Filter measurements in real-time, by averaging noises during acquisition of the frames and getting smоothed to some extend data".

Sorry, I don't understand what you are referring to here. What sort of filter, spatial or temporal? Can you give me an example of applying such a filter "during acquisition of the frames"? As I cannot see how one can interact with CCD/CMOS data in mid-exposure.

If we know that the star doesn't change brightness so quickly, there are no need to use short exposures and we can use long exposures.

Yes, there is no need to sample quickly, if the inherent variability of the star itself is slow. But the sampling rate really doesn't matter in this case: data sampled quickly can be simply re-binned (coadded), post-exposure, to the equivalent of slower sampling.

There is only a disadvantage in temporal over-sampling if the camera has high readout noise in comparison to the Poisson noise of the star and sky background.

BTW, I wouldn't use the term "filtering" to describe light curve smoothness which is simply due to acquiring more signal per data point.

I have read that it is usefull to defocus the star a little in order to get good profile. I am not sure, but this can make choosing the aperture for the photometry more stable.

Yes, there is that advantage on the aperture stability. Another justification is that very bright stars, which would saturate if in focus, can be measured. But defocusing has the disadvantage of reducing signal to noise. If the defocused, spread-out star is not bright with respect to the sky background noise and readout noise, the losses of this technique can exceed the gains.

There is only a disadvantage in temporal over-sampling if the camera has high readout noise in comparison to the Poisson noise of the star and sky background.

I should have added: "...if the camera has long readout time and high readout noise..." , because no data can be taken during the readout time.

In a camera with frame-transfer capability, such the L3-CCD in GUFI (https://www.researchgate.net/project/GUFI-Galway-Ultra-Fast-Imager ), the "dead time" is only a few milliseconds.