# Equations for Milankovitch Factors?

Are there equations for the radiative forcings of the three Milankovitch factors at any given time? Or a database of values?

There are many models that estimate orbital parameters at long timescales. All of them rely in rather complicated formulas that account for the perturbations generated by other planets, the Moon, and many other factors. Some of the models even consider relativistic effects as described in this answer.

An orbital solution that I often see cited and used in papers dealing with climatic forcing is LA93, computed by Laskar et al. (1993) and described in the paper Orbital, precessional, and insolation quantities for the Earth from -20 MYR to +10 MYR. However, more accurate solutions are available by now, like LA2004 and more notably LA2010. I'll describe below how to download output data and codes for each of these solutions.

For LA93, the codes (in fortran) and outputs of that model are available at the VizieR catalogue, where you can query the model output, or just go into this FTP folder and download the files for the code and long tables with the model outputs every one thousand years.

If by "the three Milankovitch factors" you mean Obliquity, Eccentricity, and Precession index (Eccentricity $$\times$$ sin(longitude of perihelion)). That data is ready to download and use in these two files:

The byte-per-byte description of these files as described in the README is

--------------------------------------------------------------------------------
Bytes Format  Units   Label     Explanations
--------------------------------------------------------------------------------
4- 11  F8.0   1000yr  t         Time from J2000 in 1000 years
13- 31  D19.12 ---     e         Eccentricity
53- 71  D19.12 ---     CP        e * sin(longitude of perihelion
from moving equinox)
--------------------------------------------------------------------------------


With the code you can also compute common climatic proxies such as isolation at 65°N or any other latitude.

For LA2004 (-50 Myr to +20 Myr) the same research team implemented a very useful online calculator that allows to compute insolation quantities derived from the orbital and precessional quantities for any latitude of interest. It outputs the results in tabular form. The calculator interface looks like this so you get an idea of the available options. The LA2010 solution is described in the paper La2010: a new orbital solution for the long-term motion of the Earth, and goes 250 Myr back in time. You can find similar data files and codes for this solution at the Astronomical Solutions for Earth Paleoclimates page. To my understanding, LA2010 represent the state-of-the-art solution for orbital parameters and it does include relativistic corrections as well.

I think Camilo Rada's answer is the one to follow if you want to do anything serious with the data.

However, to answer the question a bit more directly, if you want to use actual equations then an alternative that should work just fine if you are looking at just the Holocene (maybe all of the Quaternary?) is to use the original tables from André Berger

https://journals.ametsoc.org/view/journals/atsc/35/12/1520-0469_1978_035_2362_ltvodi_2_0_co_2.xml

The equations are described at the beginning and "decompose" the orbital mechanics into a sums of sinusoids:

ε = ε* + ∑ Aicos(fit + δi) e = e0 + ∑ Ei cos(λit + φi) e sin( ω ) = ∑ Pi sin(αit + ζi)

These values are read in from the tables provided in the paper. If you're just playing around, the even just summing the top 3 terms for each orbital parameter will give you nice looking results.

However, as the other answer illustrates, even though the final result looks like a rather simple sinusoid, in reality is has a lot of secondary oscillating terms. You look at the tables and the first 5+ terms are of a "significant" amplitude. So the motion is not just a couple of dominant frequencies with some noise overlay.

Again, this is all fun and educational, but if you're doing serious work then you probably want to get the full Laskar tables