Whenever I see videos on YouTube they bank really sharply to the right after disconnecting from the tanker. Why wouldn't they do a more shallow 30-degree bank?
See this video for reference:
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Fighter jets are very maneuverable, so they may make anything from a shallow bank to a breakaway maneuver. The breakaway is a standard way for fighter jets to exit a formation. It provides a way to safely and quickly gain separation from the other aircraft.
In this case the bank is fairly shallow, but when the aircraft disconnects they are already in a shallow right turn, so the total bank angle is a bit larger.
Doesn't really look all that aggressive to me, either way the fighter and the tanker are very vulnerable while refueling. Usually there is more than one aircraft waiting to refuel, so the goal of this game is to run as many of the aircraft in formation through refueling as quickly as possible.
In order to do that, you need to get your wake out of the way for the next guy to get a smooth approach to the basket. As Fooot says in his answer, the pilot is using a standard "breakaway" maneuver to get out of the formation and wait for the other pilots in the flight to get the fuel and continue with the mission.
Plus you just got a full tank of fuel in the world's funnest military equipment...
Different air forces may operate with different conventions, but right from basic flight training the RAF classifies banked turns of 20 degrees as "gentle," 40 as "medium" and 60 as "steep," and the most commonly used are "medium".
Ref: The RAF Basic Flying Manual (1952 edition) - https://www.t6harvard.com/wp-content/uploads/2017/11/Chapter-3.pdf page 26-27.
The video is just a 40 degree banked turn.
The aircraft have no problem handling the G loads in 60 degree banked turns, and there are not going to be any complaints from passengers in Business Class that the flight crew spilled their gin and tonic!
They do this because there are 3 main parts to an air to air refuel:
After the refuel, they will bank the right to go to the reform area and will wait for the rest of there squadron. Once everyone is refueled then they break from the tanker.
A jet joins on one side of the tanker then when its turn comes moves behind to tank then when finished, clears off to the other side. Bear in mind a whole formation may be tanking and need to flow through the same process. The turn after tanking is not a hard turn in fast jet terms.
Others seem to have answered the why such a sharp maneuver part of the question, so I will answer the why to the right part. Collision avoidance in aircraft is always by turning to the right when possible. Keeping the formation in such a way that the pilot can always escape to the right is probably done to ensure safety, as it lines up with the engrained training.
Two aircraft must either be flying in formation, or be far apart, to be safe. While the fighter is in the process of leaving the tanker, it is neither. It therefore moves away quickly using the standard breakaway manoeuvre. Bank really sharply? The video showed a positively leisurely turn in the circumstances.
Why to the right? It's standard to approach on the left side of the tanker. That's where other fighters waiting and approaching to refuel will be, so the left is a good side to avoid.