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I'm following along with Dr Kelso's columns on CelesTrack and I've come across this equation.

$$θ_g(0^h)=24110^s.54841+8640184^s.812866T_u+0^s.093104T_u^2- 6.2\times10^{-6} T_u^3$$

As you can see, there are superscript $^s$s right before the decimal point on some of these values and a superscript $^h$s next to the 0 argument in the function definition.

What do these signify? Is it the unit of the terms? If so, it's weird notation I've never seen before.

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    $\begingroup$ The " Explanatory Supplement to the Astronomical Almanac" is available here archive.org/details/… where you can read the page 50 reference. The whole "Time" chapter seems to use that notation for days, hours, minutes, seconds. See page 41. $\endgroup$ Jan 10, 2023 at 14:42
  • $\begingroup$ Perfect, thank you! Do you want to write that up as an answer or should I? $\endgroup$ Jan 10, 2023 at 16:26
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    $\begingroup$ Go for it, not my wheelhouse. $\endgroup$ Jan 10, 2023 at 16:35
  • $\begingroup$ FYI StackExchange supports LaTeX markup via MathJax, which you could use instead of the inline image. $\endgroup$
    – Erin Anne
    Jan 10, 2023 at 19:28
  • $\begingroup$ @ErinAnne - Ahh, I thought that was only enabled for Math.SE $\endgroup$ Jan 10, 2023 at 21:17

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$$θ_g(0^h)=24110^s.54841+8640184^s.812866T_u+0^s.093104T_u^2- 6.2\times10^{-6} T_u^3$$

What do these superscript "s"s signify? Is it the unit of the terms?

In this expression, the superscript "s" represents mean hour angle sidereal seconds. This is a measure of angle. There are 24 hour angles in a 360° rotation, so 24*60=1440 hour angle minutes in a 360° rotation and 24*60*60=86400 hour angle seconds in a 360° rotation. Hour angle is handy way of representing rotation because it corresponds nicely with time. Just as there are 86400 UT1 seconds in a mean solar day, there are 86400 mean sidereal seconds in a sidereal day.

Note that the referenced expression is valid at midnight UT1 only as the expression omits the 360° (or 86400 hour angle seconds) that the Earth rotates through per day.

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    $\begingroup$ To convert hour angle seconds to arcseconds, multiply by 15. To convert hour angle seconds to degrees, divide by 240. $\endgroup$ Jan 11, 2023 at 12:36
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    $\begingroup$ It's really weird how it's put before the decimal point rather than after the number $\endgroup$ Jan 11, 2023 at 14:14
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    $\begingroup$ @ScottishTapWater This is fairly common in astronomy. For example 123.456 degrees will oftentimes be marked as $123^\circ.456$ as opposed to $123.456^\circ$. $\endgroup$ Jan 11, 2023 at 14:21
  • $\begingroup$ How strange... Although I must say, it's actually arguably less ambiguous because there's no chance of confusing units with algebraic terms $\endgroup$ Jan 11, 2023 at 15:15

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