Since celestial objects move along the sky, taking long exposure shots with an unmotorized telescope will show a straight or slightly curved line of light instead of the still object (star trail).
Motorized scopes don't have this problem because they can be set to follow the rotation of the earth or objects in the sky. Unmotorized scopes can't do this, so long exposures are not practical. What is the optimal exposure time for photography with an unmotorized telescope?
Motorized scopes don't have this problem...
Not necessarily!
Depending on the type of mount, they might have the second order problem of field rotation. For properly aligned equatorial mounts the telescope tube is also rotated about its axis when it is rotated around the polar axis of the mount, so for example north is always pointing in the same direction at the focal plane of your camera.
For telescopes on alt-az or "azimuth" mounts this doesn't happen, so on the scale of hours any object that's not in the center of the field of view of a properly aligned telescope will execute a circular arc around the center of the exposed field.
What is the optimal exposure time for photography with an unmotorized telescope?
It depends on what you define as "exposure time". There's total integrated time used to produce your image which can be quite long, and your frame rate, how frequently you dump one "exposure" to memory and start recording a new one, for later recombination into a single final image.
Most astrophotography both pro and amateur relies on the latter technique to avoid overexposing bright objects in the field that might only need a few seconds or minutes to saturate, while still capturing dim objects.
Saturation can cause some CCDs to produce other artifacts in other areas of the frame, so you really do want to avoid it.
So if you don't have tracking, you may need to make your frame rate of the order of one per second or even less, depending on the scale and resolution. I think other answers here will go into more detail on that, or at least link to other questions and answers here that already address this.
Note: Once you are making many shorter exposures, you can then pause to move your object back to keep them near the center of the field at regular intervals. The software you use to combine images will do the realignment as long as there is not appreciable field distortion. If there is, then you'll need to work harder to keep your object centered, or include a complicated model for it in your image-combining software.
Modern DLSRs, astrophotography cameras and even some cell phone models will have ways to program regular frame dumps for image accumulation.
There are numerous programs you can buy or get for free to reassemble those images into a single final image. These automatically align all of your exposures before adding them, and have features to lower the resulting noise.
To prevent star trailing, there are two different methods in general use. One is the 500 Rule which can be done in your head, or quickly with a calculator. The other is the NPF rule which is more accurate, but generally requires a program on hand, but many can be found with a simple search.
500 Rule
$$ t=\frac{500}{f} $$
$ t $ is the time in seconds of the exposure, and $ f $ is the focal length of the telescope/lens. It is common for other numbers such as 600 or 400 to be used. This is just a general rule of thumb, and results will vary depending on the resolution the final result will be displayed at.
NPF Rule $$ t=k \frac{16.9N + 0.1f + 13.7p}{f \cos\delta} $$
$ t $ is the exposure time in seconds, $k$ is the acceptable number of pixels of trailing, $N$ is the focal ratio (f-stop), $f$ is the focal length (in mm), $p$ is the pixel size (in microns), and $\delta$ is the declination of the object/area.
To compute $p$, you will need to look up the size of your sensor and it's resolution. Then compute $ p = \frac{sensor\ width\ in\ microns}{image\ width\ in\ pixels} $
For a wide angle image, use the declination closest to the celestial equator within the field of view.
