Is it possible to create a picture like the following, using a D7100 camera with a standard 55-105mm lens. If not, what is needed for this kind of photography?
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Those two photos are heavily processed - you can't get that all in one image with one exposure. I'm not saying it's impossible, just extremely unlikely, from looking at the images. In fact, in the top one, she didn't do the reflection correctly, and the reflection doesn't match what it's reflecting - it's not obvious but there's some large errors in the reflection that don't look natural. This is a fact of photographs like this - you need to learn your lightroom or you won't realize this combination.
You can however, take some pretty amazing starlight shots with your current equipment. You will need to experiment and learn how to do it, and you'll probably spend several hours in the dark before you get one right. To replicate the example you would need a wide angle, but I've used zoom at night and liked it. The stars are so far away it's not going to change the look of the photo, only how much stuff is in it. This is one case where you can 'compose with zoom' and it's all good. Keep in mind, photos of only stars don't look so good - that's why the above examples are so striking - they have good framing.
Make sure you know how to use all the controls on your camera and you can find them in the dark - turn off the screen, it's too bright! I agree this is a really general question and it's hard to give a complete answer. So, your best bet is going to be the web...
Google can find a ton of articles about this - it's pretty popular: https://www.google.com/search?q=how+to+photograph+the+stars
Here's a pretty good article I just found too, which has a good discussion of how to process a single image, and it shows the original and what it looks like after each step, so you can really see the difference.
I would say it depends what kind of photo you are taking. If it's the Milky Way with a background/foreground in it, you have to go for a faster lens with a wider focal length like 21mm or below. And that too you are limited to 25 seconds of exposure as the earth rotates and beyond this amount the stars are going to trail. In your case you have to divide 600 by your focal length * the crop factor
So if you are shooting at 55mm your total exposure time will be 600/(55*1.3) which is about 8 seconds. That's not enough to capture the milky way, you will need at least 20 seconds for good results.
You also need a pretty high ISO which will deliver acceptable results. For example I normally shoot the Milky Way at iso3200 on my 6d, I can go to iso6400 but it gets noisy so too high of an ISO can be worse.
You also need to shoot in a light pollution free environment as light tends to go orange and since the Milky Way has orange parts in it, during post processing while you attempt to remove light pollution it will affect the orange gases in the Milky Way as well.
If you want to shoot deep space objects like nebulae you will need a tracking mount as some of the exposures can take several minutes. You could use a tracking mount if you just want to shoot the Milky Way without any foreground objects else during tracking the foreground objects will trail. Bonus is if you use a tracking mount like astrotrac, you can shoot multiple shots of the Milky Way at lower ISO and then stack them using dpp.
Yes, but it's not easy. Lens and camera are far less important than how you mount your camera and process your images - astrophotography is too involved to address in a single answer on stackexchange, but in short you need to mount your camera so that it follows the motion of the stars across the sky, and composite many exposures to increase your signal:noise ratio.
Google for "tracking mounts", "barn door trackers", and "image stacking for astrophotography", and you'll be able to start your (long) journey towards producing that sort of image.
tl;dr - you need patience by the ton.
Actually contrary to what some are saying here, I have found astrophotography is not nearly as difficult as many advocate and you do not need a tracking mount. Astrophotography can be done with a crop frame sensor, but you are typically limited to iso's ranging from 1,600 to 3,200 and ideally some of the best photos come at iso's ranging from 3,200 to 6,400, which requires a full frame sensor to capture this with little noise. I would recommend a good 14mm and 24mm lens with focal ratios ranging from 1.4 to 2.8 and nothing larger. The exposure length will depend on the focal ratio, but ranging from 50mm to 14mm I find the exposures vary from 10 seconds to 25 seconds and the shorted the exposure the higher the iso must be to compensate. In short you need a sturdy tripod, a remote trigger or an intervelometer, preferably a full frame camera and a low focal ratio (1.4 to 2.8), short focal ratio (14 to 24mm) lens.