How would a planet need to be changed (gravity, air pressure, air makeup, etc) to allow for the realistic flight and take off from a small hill of a gryphon with a human rider? You can assume the gryphon has hollow bones all through its body, but what other changes would be beneficial? I am asking for changes to a planet that would allow for this, rather than merely asking if it is feasible on planet earth.
I believe you would want atmospheric conditions similar to earth's during the Cretaceous period, with a cooler global climate. I will first explain why I think this correct. Then I will describe the relevant differences between the optimal atmosphere and today's atmosphere. Then I will cite my sources.
Why: The Pterosaur Quetzalcoatlus northropi weighed around 400 to 500 pounds. According to the National Geographic article referenced below, they could have flown nonstop for 10,000 miles. In the process, they would have burned off 160 pounds of body fat. Therefore, they most have been able to get off the ground with 160 surplus pounds of weight. If you put a Quetzalcoatlus on a diet, had it lose 160 pounds before it went anywhere, and then replaced the lost body fat with 160 pounds of rider and tack, it should be able to get off the ground and fly for a little while. At this point you're riding a starving dinosaur, which comes with its own set of problems. But these can be mitigated to some extent.
These dinosaurs had the size and strength to fly while carrying a slim human. They were active during the Cretaceous period. There were some beneficial atmospheric differences that may have helped them fly.
Relevant differences in atmosphere (from Wikipedia):
The period in question was known for being warm and full of huge plants. Oxygen was 150% of modern levels. This means your griffons will breathe in more oxygen with every breath, making them more efficient. It will also make the air more dense.
Here is a quote from businessaircraftcenter.com: "engines run more efficiently in cold air because colder air allows the engine to use a greater mass of air/fuel mixture in the same intake volume. That translates into more power."
Also: "In propeller-driven planes, the prop is biting into denser air, and thrusting a greater mass of air backwards, which means more thrust and power."
Griffons of course aren't propeller-driven planes, but they need oxygen to energize the blood and they generate lift by flapping. I think griffons would benefit from the same conditions.
Therefore, I believe a griffon would have an easier time getting airborn with a rider in "dense air" conditions. Increasing the oxygen by 50% and lowering the ambient temperature to a around freezing are two ways to achieve this. Your griffon will spend a lot more energy getting off the ground, as every wing flap is pushing through a denser atmosphere. But the tradeoff is more buoyancy, which is a worthy tradeoff.
Bibliography: National Geographic: http://news.nationalgeographic.com/news/2010/10/101015-science-giant-pterosaurs-longest-nonstop-flight-distance-record/
This question and the answers, though coming at it from a different angle, might be helpful How to make a viable flying mount?
The answer here talks about the stumbling blocks of an earth-like or earth planet for such a creature to be a flying mount. These include gravity and weight of the rider, as well as evolution favoring animals who carry more than their own weight (I think a breeding program would take care of that).
The accepted answer though had some interesting fixes on who could ride and what the animal might be like:
It could be possible, maybe, if you're willing to make some concessions, such as having only very light riders (a caste of messenger midgets?), accepting that the animal will be fragile and die easily so it can only be used for very very important purposes, and/or somehow managing to enhance the animal using nanotech, for example, to make extremely thin bones extra-strong or to increase metabolic efficiency (this could be hand-waved by implying an advanced civilization tinkered with them ages ago and then left).
But you are not asking how the animal might be adjusted, but how the environment itself might be.
The biggest one would be less gravity, of course. But add to that--smaller humans. Yes. That's one way you could change the planet--humans are smaller in size to begin with, on average, with only some who are the lightest of those allowed to be gryphon riders. Think jockeys--they are only allowed to be a certain weight for max speed and durability of thoroughbred racehorses.
Changing air pressure doesn't seem advisable--I think going either way is going to have drawbacks for you.
More oxygen at higher heights might be good because the gryphons won't tire as much.
I am also going to drop you this link on different gryphon designs from the anatomically correct series. In it, you'll find addressed the problems with flight, let alone adding a rider.
None! Such an animal could, even on earth, take upon a human rider. Take the crown eagle for instance. It can carry the heaviest external weight among all birds. They weigh about 3.64 pounds and have a wingspan up to 1.81 meters. This bird feeds upon the largest species of monkeys in the world (the Mandrill) which can weigh up to 37 kilograms. If you give your gryphon the proper biological tools and size, it could quite realistically carry a human on earth. However if you would like the gryphon to be able to carry a rider for long long distances, it might be good to increase oxygen content and decrease gravity. Decreasing air pressure too much would not give the bird enough air to push down upon and too much air pressure would burden the bird too much so I would not tamper with air pressure.
The planet needs to be have the mass and atmospheric density of the Saturnian moon Titan. The combination of lower gravity and a denser atmosphere would enable a gryphon-like creature to be able to fly and, possibly, carry a human mount.
Of course, you wouldn't want to have your gryphon flying around in a truly Titan-like planet. The human rider would need to be wearing a spacesuit. Titanian surface conditions are not very conducive to human survival. If the gryphon is a native lifeform it won't have to worry about its survival.
A planet that has the characteristics of a Titan and yet is Earthlike may be difficult to determine where in the universe it might be. Basically this means in relation to its primary star. It needs to be cool enough to maintain a dense atmosphere and yet comfortable enough temperature-wise to be sufficiently Earthlike. Mars which has a gravity similar of Titan has lost most of its atmosphere, so keeping a dense atmosphere will be hard. At the very least, the planet will need a strong geomagnetic field to prevent soalr winds from evaporating its atmosphere.
While a gryphon-friendly planet is imaginable, it is extremely difficult to conceptualize where such a planet could be located, astronomically speaking, in its planetary system. So, it's conceptually plausible but may be unrealistic in the practical sense of nature. On the other hand, there is an absolute plenitude of weird and wonderful exoplanets. Perhaps, a suitable planet Gryphon is among them.
It is, obviously, nonsense to state that increasing air pressure would reduce maximum load carrying capacity. That is, it is obvious to anyone who has ever heard of a fish. It's no coincidence that aerodynamics and hydrodynamics share most of their mathematics and models. They both describe fluids and objects moving through fluids. (as well as fluid mixing and turbulence and other nonsense). Based on the size of prehistoric fliers, increasing oxygen concentration also would be a significant advantage. And it's just as obvious that decreased gravity (less dense planet) would help as well. Two other possibilities, but I can't do the modeling: increasing the planet's spin (leading to higher Coriolis forces and far more importantly higher wind speeds) and being tilted (possibly closer?) with respect to its star. Again higher average wind speeds. The false (afaik) claim that a Crown Eagle can grab and lift a 35 kg monkey is risible. A 3 kg monkey? Sure, no problem. I point out the fairly large body of work on man-powered flight. There's some lessons to learn there. Although the fact that these type creatures don't exist on Earth should be a clue that their long-term survival is unlikely. These creatures would be quite delicate, imho, and wouldn't be able to bear a human load on the ground (at least not except for momentarily). I'd also suggest the possibility of inflatable bladders, but you were asking about environmental conditions, not the animal's design.