tolkiens legendarium - Where is the equator in Middle-earth?

In Unfinished Tales, the Essay on the Istari

Aragorn claims to have penetrated the `far contries of Rhun and Harad where the stars are strange'.

and then links to the appendix, item #10:

The 'strange stars' apply strictly only to the Harad, and must mean that Aragorn travelled or voyaged some distance into the southern hemisphere.

So where is the equator in Middle-earth? Certainly it must be south of the Harnen river and be off the canonical map(s).

Answer

In Letter 294 Tolkien describes his world as follows:

The action of the story takes place in the North-west of 'Middle-earth', equivalent in latitude to the coastlands of Europe and the north shores of the Mediterranean. But this is not a purely 'Nordic' area in any sense. If Hobbiton and Rivendell are taken (as intended) to be at about the latitude of Oxford, then Minas Tirith, 600 miles south, is at about the latitude of Florence. The Mouths of Anduin and the ancient city of Pelargir are at about the latitude of ancient Troy.

Based on this we can overlay a map of Middle-earth on a map of our world and get something like the following:

(Image credit: http://gisninja.blogspot.com/2012/12/georeferening-middle-earth.html)

This only gets us about half the distance to the Equator, which is a good deal further off the map.

Obviously descriptions are relative: "Far Harad" may have seemed quite far indeed to a Hobbit map-maker, but there's a LOT of Harad beyond even that, and "Rhûn" barely even begins the journey East.