classical mechanics - Catenary curve under non-uniform gravitational field

If one uses the functions $f_d(t) = (cos (d t))^{frac 1d}$ (which were implicitly introduced by Colin Maclaurin) one can compute the catenaries for central gravitational fields which are proportional to a power of the distance to the Sun. These are curves of the form $rf(theta)=1$ where $f$ is a function of the above type (or a dilation and rotation thereof).

Interestingly, one can also get catenaries for parallel forces from the same functions. In fact, parametrised curves of the form $(F(t),f(t))$ where $f$ is one of the above functions and $F$ is a primitive, give the catenaries for gravitational fields which are proportional to powers of the distance to the $X$-axis. These curves, the Maclaurin catenaries, are new.

Not relevant to the question but worth mentioning is that if one rotates them about the $X$-axis, they provide examples of surfaces for which all six parameters in the fundamental forms ($E$, $F$, $G$, $L$, $M$, and $N$) are proportional to powers of the above distance.

Details in arXiv 1102.1579