Q:

Why are my sky photographs coming out dark?

When I take pictures of objects in the sky (aircraft, birds, etc.), I frequently find that the entire sky looks underexposed.   I typically take such pictures in Program mode ("P"), because there is often not a lot of time between hearing or spotting the object in question, rushing to get my camera, and aiming the camera to take a few shots before the opportunity passes.  There is not a lot of time to fiddle with camera settings or make test exposures.  I shoot through the lens and not with "live" mode and don't have any feature like a live histogram available.
I have seen this behavior with a variety of shooting angles relative to the sun. I would expect that as aim closer to the sun, any object I am photographing would become more backlit and in its own shadow and thus appear dark.  But the entire sky seems dark, and I see this even when shooting with the sun at my back.  In the pictures below you can see the parts of the aircraft facing the sun and in the shade but the sun lit side seems very dark to me in the "normal" (+0EV) exposure.   I mainly this phenomena at telephoto lengths, making me suspect the lens, but since metering is through the lens I would expect that this shouldn't make a difference, at least not this much.
Typically, adding two stops to the exposure makes the shot better match my expectations.   To me this seems like a lot, like the metering system is seriously misreading the scene, yet the scene does not seem that complicated. But am I expecting the wrong thing?  Am I just expecting the picture to be too bright?  Is there some obscure setting I am missing?
Here is an example. Both images shot on a tripod around 2p.m. on a sunny day at ISO 100 f/6.3 with a Nikon D5100, a Tamron 18-270mm f3.5-6.3 VC len at 270mm (VC off) as part of an exposure bracket (-2EV, 0EV, 2EV).   Active D-Lighting is set to the highest setting.  (This setting for Active D-Lighting has shown the brightest sky images in all my tests.)  No other in camera adjustments.  These pictures were taken with "multi-segment" metering, which if I understand correct is looking at a large part of the sky and so shouldn't be underexposing the whole picture.
1/1250s, +0EV, full image scaled to 25%

1/1250s, +0EV, cropped image at full resolution

1/250s, +2EV, cropped image at full resolution

My understanding is that for security purposes the meta-data may be stripped by SE sites, but I can post any additional metadata needed (there are over 200 individual pieces, so I'm not posting it unnecessarily.)

A:

This is usually caused by the camera taking the scene as a whole and treating it as 18% gray. This is the way light meters (and camera meters) are calibrated to see the world. Scene mostly snow? 18% gray. Black cat in a coal mine? 18% gray. The camera doesn't know what you're taking a picture of, so it assumes that the scene is, on average, 18% gray and exposes accordingly. Your adjustment using the +-Ev feature is what we do to tell the camera what the real world looks like. Snow? +2Ev. Black cat? -2Ev. Here is a more detailed discussion of exposure compensation and how, why, and when to use it.

A:

Your camera's light meter measures brightness, but it can't tell if the brightness level it is measuring is a black cat in a coal mine or a white cat in a snowstorm.¹ It assumes everything you point your camera at is somewhere about halfway in between those extremes.
¹ Sure, the scene with the white cat will probably be brighter than the scene with the black cat (unless our blizzard is at midnight on a moonless night somewhere out in the boonies where there is no light pollution or maybe our coal mine is very brightly lit due to OSHA regulations), but the camera usually can't tell if that cat and its background is supposed to be black or white.
Unless you tell it otherwise, many cameras will try to expose whatever you point it at to that medium value.
Light meters have gotten a little more sophisticated in recent years, but you have to give them something to work with. When almost the entire field of view is more or less the same color and brightness, such as the sky in your examples, the added logic doesn't have much to go on. It's going to attempt to expose the sky as "medium brightness."
Some cameras are getting pretty good at guessing better with actual scenes, especially those with RGB+IR light meters which use all three primary colors plus near-infrared to meter the scene and compare it to a library in the firmware that will probably be able to tell the difference between a bright blue sky in the top of the frame and dark green forest in the lower part of the frame. The light meters in older and many entry level cameras are monochrome and can't meter in color, so they have to guess even more and often get tricky situations completely wrong.
Guidance from the photographer can go a long way, sometimes even when the photographer isn't necessarily very knowledgeable about the intricacies of exposure. One way a novice photographer can give the camera a hint is with scene modes. Most entry level cameras have a few or more scene modes.
Scene modes are a way for the less knowledgeable or less experienced photographer to tell the camera in what conditions the photo is being shot so that the camera can use the appropriate settings to maximize the chances of a successful photo.
One classic example: Snow or beach scene.
The more experienced photographer understands that a camera doesn't know if we are metering a black cat in a coal mine or a white cat in a blizzard. The more experienced photographer knows how to alter the camera's settings to make the scene look bright without totally overexposing the image or look dark without totally underexposing the image. The novice does not usually know they need to do this, much less how to do this.
Unless we tell the camera to do differently, the camera will try and make everything a medium brightness. So if the camera is set on full "Auto", a picture of a bright, sunny beach (or mostly empty sky) will underexpose a small, darker object on that beach (or in that sky) because the camera will expose the majority of the scene as medium bright!
"Snow/Beach" scene mode to the rescue!
We don't have to know how to adjust exposure for snow or bright sand at the beach, we just have to know to tell the camera we're taking a picture of a very bright scene by turning the mode dial to "Snow" and the programming in the camera will do the rest!
The same is true of the many other scene modes. It gives the less knowledgeable photographer a way to tell the camera what kind of scene they are shooting and the camera will attempt to pick the best combination of shutter duration, aperture, and ISO to use for that particular kind of scene. The photographer doesn't really need to know what the camera does to get there. They just need to be able to recognize the difference between a bright sunny day at the beach (Snow/Beach scene mode) and a night out on the town (Night portrait scene mode). They just need to be able to tell the camera they are shooting a running subject (Sports scene mode) or a static nature scene (Landscape scene mode). This allows the camera to emphasize what is most important for a particular type of shot. If conditions are less than ideal, the camera will use one of the other, less important factors for a particular type of photo to compromise and keep the most important thing as optimal as possible.
As a photographer begins to advance their knowledge and skill level, they learn how to use exposure compensation to tell the camera about the scene in front of them. Eventually they learn about different metering patterns and when each is most useful and how each affects what their camera's meter is telling them.
For example, rather than using positive exposure compensation with matrix/evaluative metering to brighten the sky, one could use spot metering to tell the camera to only meter a small portion of the field of view at the center of the scene. In the case of a black helicopter or airplane, for example, spot metering may actually require negative EC to prevent the black aircraft from being exposed as medium gray!
Notice, though, that due to the speed of the helo's motion, when your camera was at 1/250 seconds to expose more brightly the helicopter was blurrier than in the 1/1250 shots. In such a situation many photographers might consider intentionally underexposing by a bit, saving raw image files, and bumping up the exposure in raw conversion with relatively little penalty when compared to the disadvantages of trying to raise the brightness of an underexposed JPEG.
Ultimately, most experienced plane/bird in flight/airshow shooters wind up using manual exposure mode. One trick is to meter planes on the ground that are the same color and receiving the same light as the planes in the air. Another is to push the highlights in the sky to right at the edge of overexposure. Metering is followed by examining the histograms of test shots and adjusting if necessary. This works well on a clear day but may give mixed results if clouds are moving in and out of the sky while one is shooting.
There are a few manual exposure snobs that think shooting in any exposure mode other than manual exposure mode is unprofessional. I am not one of them. There is certainly a time and place when manual exposure is the best choice, but there are also other situation when other exposure modes may be better suited to getting the results one wants. Use what works for you in each situation that you find yourself shooting.

A:

The others have explained why the metering is causing issues, but I would like to add that active D-lighting actually underexposes the image in order to preserve highlights (~1 stop at max), and then processes the data/jpeg to restore midtones/shadows... IMO, it is not really helping you.
Personally, with the D5100 I would be shooting in manual mode for this subject/situation (in order to force lower shutter speeds) with auto ISO and matrix or CW metering (resulting in underexposure). And I would be recording raw files.
With the D5100 there is essentially no penalty to recording darker raw file images as long as it is done by using lower ISO's... The camera is very nearly entirely ISO invariant; so when the image is recovered in post it will look essentially identical to having used a higher/correct ISO. Except you can do selective recovery and retain the highlights if you want.
http://photonstophotos.net/Charts/PDR_Shadow.htm#Nikon%20D5100

