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The Earth's rotation is gradually slowing down. It was significantly shorter in the prehistoric past, but now we need to add a leap second every few years and we've not yet had to take any back. In fifty thousand years, it is thought we'd need a leap second every day.

The slowing down isn't constant, however. The earth speeds up occasionally and we don't need to add leap seconds quite so often.

When was the longest solar day on record?

I suspect the answer is sometime during the 1970s when leap seconds were a yearly occurrence. It is also possible I've made an error of understanding.

(If your answer is the Summer Solstice or the 25 hour DST day, that's not the type of day I'm asking about.)

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Maybe I'm reading too much into the question, but there's a few ways to measure a day. I think 1912 or 1970 is an accounting answer more than the actual length of the day, similar to how 1751 in England was the shortest year, 282 days, because of a calendar adjustment.

Earth takes about 23 hours, 56 minutes and 4 seconds to complete a full rotation and that number is very consistent but gradually increasing because of the Moon and tidal effects. It's increasing about 23 milliseconds per century or about a second ever 43,000 years or a minute every 2.6 million years. It also works out to about 230 microseconds per year, less than 1 microsecond per day.

So, in a sense, if you're talking sidereal day, each day becomes the new longest on record, by a tiny bit, unless there's a large Earthquake which can speed up the Earth's rotation a little. The 8.9 Richter scale Earthquake in Japan compacted Earth slightly and sped up Earth's rotation by about 1.8 microseconds, or slightly more than 2 days worth of tidal slowing.

The longest solar day would be perihelion day, when the Earth moves fastest around the Sun and a single rotation has to move that much further to get the sun back to where it started. According to this article, obliquity plays a role too, which I'm having a hard time picturing, so I can't explain that.

This raises an interesting problem because Earth's eccentricity is currently decreasing and has been for about 15,000 years. Earth's eccentricity was considerably higher at times in the past, perhaps a hundred or two hundred thousand years ago, perhaps as high as .04 or .05, much higher than the current .0167, which could have added perhaps 15-20 seconds to a solar day. The longest day would probably be during a recent peak eccentricity during perihelion. Unless I'm overlooking something. Corrections welcome.

Eccentricity Wikipedia article 1

Eccentricity Wikipedia article 2

enter image description here

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I think the question is very interesting, but a bit difficult to answer due to a lack of detail and accuracy in data dating back from pre-atomic clock era. Also, we have to decide which type to day to analyze, mean solar day or apparent solar day? From the elements in your question, you seem to be describing the mean solar day, so let's analyze this one.

One thing that we can notice in several studies, such as this study by Morrison and Stephenson, is that the length of day (LOD) was longer on average around 1900-1915, with an average day length close to 4 ms longer than 86,400 seconds:

enter image description here

Before 1900, the tendency is really towards shorter day values, so I think it is logical to assume that the beginning of the 20th century saw the longest day values (at least on average), and that the longest day likely occured somewhere in that period, although I can't confirm this, nor can I confirm which day it was, because LOD fluctuates quite a bit, and data from occultations, transits and eclipses back then just aren't detailed nor accurate enough for such an analysis.

If we analyse the time period where daily data is available with good accuracy (approx. the last 60 years), on the IERS website, then based on this data, which includes daily values of LOD since 1962, and taking into account the margin of error, I think that October 31, 1971 and April 12, 1972 are the two contenders for the longest day for that period (highest LOD value between 1962 and today), with 0.004348s and 0.004355s respectively.

Here is the graph of LOD from the IERS website: enter image description here

In the future, as you know, the days are expected to get longer and longer, so eventually, the record should be broken. How soon is anyone's guess...

Analyzing the apparent solar day would be much more complex due to several factors that affect the actual apparent length of the solar day, like the position of the Sun, planets, orbital eccentricity and axial tilt.

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