What is a powerful good handgun to take trail ridding?

taurus raging judge in 454 

For putting down a horse in an emergency situation a knife is sufficient. You should be able to carry an adequate knife even in a National Forest without any problem. Basic woodsman equipment. If anyone asks, "It's for cutting my horse free if he gets tangled in a rope." I never go anywhere without a knife.

The plan is to get both carotid arteries. Yes, it's a bit messier than an ears-eyes put-down shot, but it will work and is not dangerous unless the horse is thrashing severely.

A water gun. It scares away all predators. 

While I understand what you are asking, Kat, there are a few things I should point out. One of which is that you CANNOT effectively kill ANY injured horse from a distance. I know, you've likely seen a lot of movies where the sheriff or some other law enforcement officer- or a rancher or some other character does that with ease, but that's just Hollywood fantasy, and you need to understand that in real life, it never works that way.

In order to kill any horse quickly and humanely with a gun, you need to have the barrel of the weapon positioned in the correct place, right against the horse's skull. Because otherwise, you risk just simply adding insult to the already existing injury, and causing the animal even more distress. Even worse, shock, stress, and the emotions of the situation will have a negative impact on your ability to shoot correctly, which is something that you aren't taking into account. This is why they don't use guns to put horses down on the racetracks, Kat. Racehorses who break down catastrophically are always euthanized using chemical means.

Another thing you probably don't realize is that many times when a horse dies suddenly by a bullet, the animal will continue to thrash for several minutes, even though he or she is already dead. That's because of the sudden disruption in the central nervous system that the shooting causes. This is upsetting for anyone to watch, which again is another reason why guns aren't normally used to euthanize horses in public situations. Horses are not cattle, Kat, and can't be shot the same way that cattle are. Captive bolt guns- like the kind that are used in slaughterhouses that cattle are sent to- are USELESS when it comes to horses, because of the differences in the structure of the animals' skulls. One of the most horrendous things about the slaughterhouses in Mexico that do process horses is that they often use this type of gun- with the result that many horses are still ALIVE when they are strung up on hooks so that their carcasses can bleed out. Have you ever heard a horse scream, Kat? And I am not talking about a stallion issuing a challenge to another stallion, either. I HAVE had that experience- and it's not a sound I will ever forget.

Do you even know the correct place to put a bullet if you want to shoot a horse, so as to end the animal's life as quickly and humanely as possible? It's not where you probably think it is. To locate it, you need to draw an imaginary line from the base of each ear across the horse's face to the corner of the opposite eye. The point where the lines cross- in the center of the forehead- is where the bullet goes. Large caliber rifles and or pistols are recommended.

First off, is your concern about running into bad people or bad animals? If it is people then use whatever gun you normally use for concealed carry, it will work just as well in the woods as in town. If it is animals then you have to define what animals. What is good for snakes is not so good for bears. Also you are unlikely to run into a grizzly in Maine but you could definitely run into a moose in a bad mood. Lastly, what can you actually handle with one hand? You may not have time to get both hands on the gun and assume a nice Weaver stance, you may have to fire one handed from the hip. So what can you handle one handed? If you get a 454 Casull or 480 Ruger you will probably be able to get off one shot and that is it. Might be better to go with a 357 or 44 mag. I carry a single action Colt 45. Sure, it takes two steps to fire it but that is like a safety in my mind, a kid might not realize they have to pull back the hammer first.

In the woods, on an ATV. The chances of you being attacked by a aggressive ferocious animal is as close to zero as you can get. Wild animals run from strange noises. Not at them. If you need a weapon. A short barreled shotgun with a barrel length of 18-22 inches is better. Preferably with a pistol grip/butt stock conversion kit. Easy to get a quick release mount and easy to use. Loaded with 00 buck. It'll stop any attack right quick. Except for a full grown bear.

Update- I just reread your question. You are NOT going to be carrying any weapon on federal land without express approval.

NO gun. Being inexperienced with guns in an emergency you're far more likely to shoot yourself or your horse.

A 357 magnum revolver is a common sidearm in the situation you are describing .  Ruger or Smith&Wesson is a good brand .   

Nothing less than a .30-06 handgun will do.