Venus' "analemma" over $8$ years

After the post on Jupiter's "analemma" over $12$ years, I wanted to do the same for Venus, an inner planet. Venus' orbital period of $224.701$ days makes it so that $13$ orbits of Venus are approximately equal to $8$ Earth years (with the error being less than a day). This means that roughly every $8$ years Venus comes back to the same approximate position in the sky - good enough to consider it a closed curve as far as its "analemma" is concerned.

Since Venus is an inner planet, its mean meridian transit period is effectively a mean solar day. I took Stellarium screenshots with a gap of $6$ days spanning the required $8$ years. This was then rendered into a movie that I've uploaded to YouTube (see below).

A bonus is that the Sun's analemma can also be seen in the video as the Sun traces out the figure-8 image near the central meridian.

Please manually select the High-Defintion (1080p) quality if YouTube serves up a lower quality version.

I manually traced out the "analemma" curve for Venus and overlaid it on the first image. The path is basically a Lissajous curve and very complex indeed. (Since the curve was manually traced based on Venus' positions in the screenshots, there are quite literally some rough edges in there.)

Venus' "analemma" over $8$ years

Jupiter's "analemma" over $12$ years

Introduction

In Oct-2026, a discussion on Reddit spurred me to think about what it would mean for a planet to have an Analemma. I shared my thoughts on 20-Oct-2026. The relevant portion of that response is reproduced verbatim below.

...

For a planetary analemma (if one can call it that), things are a little complicated. You need to capture the direction vector to the planet at a period that equals the mean duration between two successive transits across the meridian by the planet. ... 

For the outer planets (Mars onwards), to a first approximation, you can imagine that the planet is going around the Earth in an approximately elliptical orbit. This would give you a very similar analemma curve as the Sun or Moon - except that it will only be a piece of the curve and will not be a closed loop as the planet will NOT return to its original position at the end of a year. Further, there will be a retrograde loop superimposed on this main curve. How this would look like would depend on the relative positions of the planet and the Earth in their orbits.

(The inner planets are going to be even more complicated.)

But, in theory, you could take a tool like Stellarium and take screenshots periodically (i.e., at a period that matches the mean duration between two successive meridian transits for the planet you're interested in). Then overlay all those screenshots on top of each other so that the meridian is aligned in all the shots. ...

...

I eventually followed up on my thought process above by taking screenshots on Stellarium periodically (at $7\times$ the mean meridian transit period for Jupiter) covering a span of $12$ years as that subsumes Jupiter's orbital period of $11.86$ years. The screenshot sequence was then converted into a movie. I shared the movie on 22-Oct-2026.

The rest of this post details the mechanics of how I did it.

LTCG harvesting in a non-zero LTCG tax bracket

Introduction

In the context of US tax law, does it ever make sense to harvest long term capital gains (LTCG) when one's income puts one into the $15\%$ LTCG tax bracket? Perhaps there is value to be gained by paying LTCG taxes now and increasing the basis so that one pays lower LTCG taxes in the future. Is that opportunity cost worth it?

This post investigates the conditions for LTCG harvesting to yield a net benefit in after-tax spendable wealth under a simplified model. It is found that LTCG harvesting has a clear long-term advantage when done in the $0\%$ LTCG tax bracket, whereas harvesting in a positive LTCG tax bracket can provide an advantage only over a finite investment horizon assuming future LTCG tax rates are sufficiently higher. The longer the post-harvest investing duration, the greater the erosion in the advantage. Eventually, once the post-harvest investing duration crosses a break-even horizon, any advantage due to LTCG harvesting (done in a positive LTCG tax bracket) is completely lost.