How to find period/envelope that is larger than sample size? Mercury's distance from the solar system's barycenter (in kilometers) 
can be approximated as: 

5.9115960588705115e7 + 1.1573608483842954e7*Cos(2.086944367456105 - 0.0029751428085695573*t) 


based on 8 years worth of hourly "samples" from HORIZONS. (t is in 
hours; I forget what value corresponds to t=0, but it should be 
irrelevant to my question). 

The residuals after this approximation look like this: 



I could add additional Fourier coefficients, but there appears to be a 
period or envelope that is larger than the sample size. 

Standard Fourier analysis will never find this period, and continuous 
Fourier analysis is fairly inefficient. 

How can I find this period/envelope? 

Perhaps more to the point: what's the simplest function that does a 
"fairly good" job of approximating this data? 

The graph shows a pattern, so there "must be" such a function?
If you can guess the number of Fourier coefficients, then a decomposition method such as MUSIC or ESPIRIT might provide a parametric estimate or eigenvector decomposition, given that guess.