The catenary curve is the shape of a chain hanging between two equal-height poles under the influence of gravity. But the derivation of the (hyperbolic cosine) curve equation from the physics traditionally assumes a uniform gravitational field. Suppose instead one uses the non-uniform gravitational field that diminishes with distance from the center of the Earth. (Perhaps this would be relevant for a very long chain that sags significantly.) Does this lead to an interesting curve, known in some closed form? Or just to a differential equation that can only be solved numerically?
I ask this primarily out of curiosity, so please interpret in that spirit!