Astrophotography Gear ($5K Budget)
I would 1st research the AP targets you are interested in; note the wide range of sizes of these targets and make sure these are visible at your location, with altitudes of >30deg. With this info, you will find out that no one setup can fit all of these targets. For DSOs, I break down in approx. 3 sizes: Small: most galaxies (M51, 11'x6'); Medium: most nebulas (M42, 90'x60'); Large: M31 (180'x40'), North American + Pelican Nebulas (120'x120'), Cygnus Loop (180'x180'). Planets are much smaller.
Then, investigate your locations sky conditions; how dark is it? (Bortle Scale); what are the typical seeing conditions? (arc-sec, see Meteoblue.com). Seeing conditions will greatly impact how much detail you can achieve. Darker skies will make RGB imaging more doable, allow longer exposure times (before sky fog clips your histogram), and allows good images with fewer hours of integration. Urban and a lot suburban skies will make imaging in narrowband, with a mono camera and filterwheel, a necessity.
Field of View (FoV, framing composition in X by Y arc-min) and image-scale (measure of detail in arc-sec/pixel) are achieved by the combination of focal length and aperture of the telescope and sensor size and pixel pitch of the camera. Wider FoV can be achieved by a larger sensor or shorter FL scope with larger image circle. Image detail can be achieved by smaller pixels or longer scope FL; all things being equal, larger scope aperture can resolve finer detail (Dawes Limit). F-ratio (FL/aperture) determines the length of the exposure time; low f/ is "faster", allowing shorter exposures for the same light gathering.
The one thing that is important all AP targets is a good mount, though it becomes more critical and expensive for shooting your small targets, with long FL and heavy scopes and highly detailed image scales. At this budget, I recommend a SW EQ6R-Pro ($1595, $1345 Sale), a very good budget mount w/ 44# rating. To get better you would pay much more for Losmandy G11 (what I got) or iOptron CEM60 (or newer CEM70), which have higher 60# rating and lower periodic error. These are also high value mounts, but cost $3K to $4K.
For more budgetary constrained mount options, the advice to "put your money where your mount is" is very good advice. Another good rule of thumb is to keep your total load at 50% of manufacturer’s stated, though Losmandy's loads are supposedly for AP (but I would still haircut it). The normal budget picks are SW HEQ5 (or Orion Sirius twin) or iOptron CEM40, or for very light imaging loads, the iOptron CEM25. The widely available budget mount, Celestron AVX, seems to have serious problems (lack of bearings on DEC), though few have gotten “lucky” with a unit that performs OK. ES has a couple of decent low priced mounts for the seriously budgetary challenged, the EXOS2GT (PMC-8 version) and iEXOS-100. The EXOS2GT is more robust and capable of the two, and I expect is worth the very modest price difference. All of these lower cost mounts are more limited in their ability to handle longer FL scopes, heavier imaging trains, or in guiding longer exposures.
If you will only be doing AP with a light DSLR and moderate FL lens (up to 135mm), camera trackers are popular, but they are much more limited than the above mounts. Most camera trackers: 1) only move in RA (DEC is fixed); 2) very few have the ability to autoguide (and then only in RA); 3) do not have ability to GoTo; 4) do not have ability to finely tune pointing by platesolving; and 5) do not include a tripod. Think of the iEXOS-100 as a camera tracker, but without the above disadvantages, and the ability to handle a light 80mm scope and camera, all with a similar price to the more capable camera trackers. The camera trackers main advantage is that it is very light for those who want to take their light imaging rig on a hike.
You might ultimately want a couple of cameras to give you a wider range of image-scales and FoVs. But with your budget, get only one, but make sure that it makes sense for your year 1 scope and target size. If you already have a DSLR, use it at first. I think crop sensor DSLRs make more sense than full-frame, as most scopes will have problems with vignetting and field flatness with those large sensors, even with a field flattener. That said, use what you have. If you do not already have a DSLR (or have a need for one for general photography), do NOT buy one. Cooled astrocams make MUCH more sense. I would suggest getting a monochrome camera with filterwheel/filters, as they give you more flexibility for narrowband, are more efficient, and give you more resolution. I suggest a 7 or 8 position filterwheel, which allow for LRGB and NB filters. With budgetary constraints limit yourself initially with on LRGB filters.
Two of the best deals in astrocams are the ZWO 183MM-Pro ($1000; which I have) and 1600MM-Cool ($1280). The 183 has a smaller sensor (small FoV), but it is higher resolution and more efficient. The ZWO has attractive priced packages with the 1600 with filterwheel and filters. Filterwheels and filters will cost several hundred dollars more. You can save money by getting an OSC camera, but you will be giving up flexibility to shoot NB, efficiency, and resolution.
Get gear that is good for one of the AP target sizes; I would suggest start with the Medium size targets. This is NOT starting "small" as in cheaper, but start with high quality, with gear you will keep and use in the future. This will allow you to learn and get more intuition on what works and what is important. A long scope is NOT "better" than a shorter scope, as a hammer is not better than a screwdriver. The "best" scope is the scope matches your needs to shoot the particular target, and the "best" for M51 would more than likely be "terrible" for M31.
For medium size targets, the normal suggestion is high quality, APO refractors of 70-80mm aperture of f/6 or faster with a field flattener and possibly a focal reducer. This can be expanded to larger, but faster scopes, such as Skywatcher Esprit 100 (550mm FL, f/5.5), and AT92 (506mm FL, f/5.5). I have the Esprit 100, which I shoot at both 550mm (native, included FF) and 413mm (f/4.13, with TSAPORED075 FF/FR), and I love it. I highly recommend the Esprit 100 ($2500, with everything included, even FF), with the idea of later getting a FR for a wider and faster option. Other good lower priced options include the Esprit 80 (400mm FL, f/5; $1650, with everything included, even FF), WO Star71 II (350mm FL, f/4.9; $1200, petzval design, no FF necessary), WO GT71 (419mm FL, f/5.9; $828 + $198 for FF/FR), and SV SVX080T-3SV (480mm FL, f/6; $2000, with everything included, even FF). TS Optics out of Germany also has many good value scopes, and the Sharpstar 61EDPH and 76EDPH look like very intriguing budget picks, especially when paired with their FF.
When choosing your camera, estimate is FoV with your specific scopes FL. You want this FoV to be larger than the FoV of your target, to allow cushion for: stacking artifacts, dithering, differing camera rotations, and slight framing errors. My Esprit 100 with FF/FR, 413mm FL, with my ASI183MM-Pro camera gives me 110'x73' FoV with a 1.2 arc-sec/pixel image scale. At the native 550mm, this combo gives a higher resolution and tighter 82'x55' FoV with a 0.9 arc-sec/pixel image scale. The ASI183 is practically highest resolution camera with a decent size sensor available, and can be used with longer FL scopes for small targets, if seeing conditions are good enough.
You will also have extra costs for guide-scopes, guide-cam, various cables, USB hubs, dew heaters, power supplies, computer, software, etc. …