Stereo Pairs

There's a new page on the RetroMat website with some stereo pair pictures I've taken, for cross-eyed free-viewing. 

This one (hyper stereo), however, is for parallel free-viewing. It's easier on the eye, but by necessity has been reduced in size.

Estimating aphelion and perihelion for Mercury and Venus

In an earlier post, I estimated the orbital radii for Mercury and Venus, by taking the sine of the average greatest elongation angles for each, using data for 2014 to 2029 from here.

In fact, I've been treating the orbits as being near-circular, but recently I've been wondering how to find their true shapes (which we know from Kepler and Newton are ellipses). Specifically, can I use the extremes of the greatest elongation angles to find accurate figures for the furthest (aphelion) and closest (perihelion) approaches of these planets to the Sun?

So, here is a summary for the inner planets. I've separated greatest elongation observations by East and West. The sines of the minimum and maximum angles within each set of observations provide estimates of perihelion and aphelion respectively and as can be seen are nearly exactly the same as the true figures, with exceptions to the exact figures highlighted with an *. This surprises me, because I'd assumed that I would have to find and make an allowance for the Earth's orbital eccentricity (loosely, non-circularity) for greater accuracy (which I have not done yet), especially so in the case of Venus, because the Earth's orbit is actually the more eccentric of the two planets.

Notice how eccentric Mercury's orbit is!

Mercury

• Greatest Elongation West
• 50 observations
• min. 17.9 (sine: 0.31)
• max. 27.8 (sine: 0.47)
• Greatest Elongation East 
• 51 observations
• min. 18.1 (sine: 0.31)
• max. 27.4 (sine: 0.46*)
• wikipedia: perihelion 0.31 AUs, aphelion 0.47 AUs

Venus

• Greatest Elongation West
• 10 observations
• min: 45.8 (sine: 0.72)
• max 47 (sine: 0.73)
• Greatest Elongation East
• 10 observations
• min. 45.4 (sine: 0.71*)
• max. 47.2 (sine: 0.73)
• wikipedia: perihelion 0.72 AUs, aphelion 0.73 AUs