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There's two different effects, both of which are made using the Shapes plugin for Unity! Both of them could most likely be done without the plugin, though.

The gravitational pull is a quad drawn between the planet and the dog, and it's using an arctan function to determine its angle using the difference in positions. The width and opacity changes based on the distance, too. This could be done in vanilla Unity by creating a procedural mesh and modifying its vertices, and adding a custom shader/material that creates the gradient with a slider for distance.

The orbiting effect is done with an arc that changes opacity + length based on the dog's velocity, but you could definitely do the same thing with a trail effect. Let me know if you want more detail!