In a previous post on scaling I tried to explain why you cannot simply double all size measures of an animal if you want to make it bigger. Leg bones in particular should become more than twice as thick. The reason was that doubling an animal's length, height and width will make it weigh not two but eight times as much. To withstand the eightfold increase in weight the cross section of the leg bones has to become eight times larger, instead of four times, which is what you get if you merely double the diameter. The wanted diameter change is obtained by taking the square root of how much the cross section has to change. The square root of 8 is 2.82, so there we are.
All that held for altering an animal's size on one and the same planet; but what happens if the animal's size is kept the same, but it is transferred to a world with a different gravity? This is easy: the leg bones once again have to withstand the weight of the animal, so the question is how much that changes under the influence of another gravity. Weight is a force: it is the product of the gravitational acceleration g and the object's mass. From that it follows that weight is directly proportional to the value of g, and that value depends on the planet; Earth's gravity is taken as the standard, so it is at 1g. Transporting a animal of any given weight to a 2g world will double that weight, and being on a 3g world will triple it, etc. To adapt its bones to these new environments, their cross section will have to become twice as large on a 2g world, three times as large on a 3g world, etc. What that means for the diameter of the bones is not difficult to work out: for the 2g world the original diameter has to multiplied by the square root of 2, which is 1.41, and for the 3g world the value would be the square root of 3, or 1.73.
The picture above shows a mass on a cylinder. The diameter of the cylinder is just right to support the weight of the cube. The cubes are thought to stand on three different worlds, with 0.5g, 1g and 2g. The cylinders have been changed to make them right for each world, so their diameters are 0.70, 1, and 1.41.
g. Suppose that your animal will run into trouble if the bone diameter increases beyond four times, for instance because too much of the animal will be bone! The graph shows that you reach that diameter value when the size increases a bit more than two times on a 2g world, but on a 0.5g world the animal can become more than three times the original size. So you can have animals with thin bones on a heavy planet, but they just have to be rather smaller than animals with similar bones on a low-gravity planet.
Finally here is the cartoon doggie (Disneius caniformis) adapted to three different worlds. Of course, there are other things to take care of when designing life forms for worlds with different gravities. While an animal's weight changes by transporting it to another world, its mass does not, and its inertia does not either. Muscle mass will need to be changed as well, as withstanding a greater weight will require larger muscles. Leg position may have to be changed as well. The more difficult it is to counter a high weight, the more likely it is that the legs will be kept vertically, i.e., without major angles between the bones. The difficulty in question depends both on an animal's mass and on local gravity. Mind you, there is more to be said on leg bending, as well as on leg splaying; the two are not the same thing, but perhaps that is something for another post.
PS. It seems to be getting more difficult to find new interesting speculative biology projects out there that aren't well-known already. I have my eye on one, but suggestions are welcome.