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    Estimating Hour Angle of Aries
    From: Frank Reed
    Date: 2019 Dec 24, 12:23 -0800

    For the standard Polaris tables in the almanacs, one needs to calculate LHA of Aries. There are tables for this, but we really only need the value to the nearest degree or three. One way to get LHA Aries is by looking at the stars that are right on the meridian. If you see a known navigational star exactly on the meridian (above/away from the pole), look up its SHA and subtract from 360. That yields LHA Aries:
      LHA Aries = 360° - SHAmer,
    where SHAmer is your best estimate of the SHA of the meridian. You don't actually need a star exactly on the meridian. For example, you may be able to estimate visually that the meridian is halfway between Sirius and Procyon and work from there. Note: if you find a star "below the pole" directly on the meridian (e.g., for northern stars, the pointers of the Big Dipper might be seen directly under Polaris, so you would look up the SHA of Dubhe), then you subtract the SHA from 360°, and add 180°. It's also worth knowing the resulting LHA Aries (these are within two degrees of the actual numbers) for a few bright stars:
      Alpheratz 0°
      Betelgeuse 90°
      Denebola 180°
      Ras-Vega 270°
    where "Ras-Vega" is my way of remembering "halfway between Rasalhague and Vega. Note that these are not the SHAs of these stars. Rather these are the LHA Aries values when each of these stars is on the meridian. If you go outside 11:45pm local mean time tonight, you'll find Betelgeuse very close to your local meridian (either due South or due North), and that means that LHA Aries at that time is 90°.

    We can also calculate LHA Aries in our heads to the nearest degree or two. It's a bit much to remember, but if you work through a few examples, it will stick with you.

    First, GHA of Aries at known date and UT (GMT):

    • Count the number of days since Sep. 21. You get a degree per day (nearly),
    • Take back a degree for every 71 days since Sep. 21 (see †),
    • Add 15° for every hour of UT,
    • Add 1° for every 4 minutes of UT.

    For LHA Aries, you then add your longitude (where west longitudes are understood to be negative numbers).
    Example: I am about 100 miles northwest of Bermuda, longitude near 64.0°W. I want to check my latitude with a Polaris sight using one of those "Q" tables, or by direct calculation of Q. So I need LHA of Aries. The date is today, Dec 24, 2019, and the UT is 22:12. How many days? I ignore the 30/31 cycle in month days usually and just use 30.5. We're three months and three days after Sep 21 so I'll call that 30.5·3+3 or, to the nearest degree, 94. Then take back one for every 71 days (one in this case), so that's 93°. Now add 22·15 hours, which is 330 and three degrees for 12 minutes of UT totalling 333°. Add that to 93° and subtract 360. Thus GHA Aries is 333+93-360 or 66°. Finally for LHA Aries, we add the longitude (but it's west, so negative), and we get LHA Aries: 66-64 or 2°, which is just about right (see ‡).

    In this example scenario, you could also have estimated LHA Aries by looking at the sky. The meridian would be about a third of the way from Diphda to Fomalhaut. If you could estimate that with a few degrees, then you would have LHA Aries within a few degrees, and that would often be good enough for the Polaris tables.

    To extend the example a bit to give a complete navigation solution, imagine shooting a sight of Vega first. Two minutes later you shoot Polaris. Work up Polaris for latitude. Then use that latitude to calculate the hour angle from your Vega sight. Add that onto the GHA of Vega, and you've got your longitude. Position fixed. No mess, no worries, no plotting.

    Finally, estimating LHA Aries has another important use, in addition to the Polaris tables. Many navigators use Pub.249 Vol.1 as a star finder and as an excellent sight-planning tool, even if they work up sights on a calculator or using some other methodology. Every line in the Main Table of that volume is a succinct "sky simulation" --it's a mini planetarium in a line of text, listing seven key stars. And to set that paper planetarium to show your sky, you need only two parameters: latitude, which determines the north/south orientation of the celestial sphere, and LHA Aries, which determined the east/west orientation of the celestial sphere. If you can quickly estimate LHA Aries, looking up the sky for your unique conditions is easy: turn to the page for the right latitude, scan down the columns to your LHA Aries estimate. A day later from the same location at the same time? Drop down a row. Four minutes later on the same night? Also, drop down a row. Two days later but twelve minutes earlier from the same latitude?? That's down two rows and up three. Just that easy.

    Frank Reed

    † Why 71 days? Because the Earth's orbit around the Sun covers 360° in 365.242 days which amounts to just about 0.986° per day. One can either take the total number of days and multiply by 0.986 or we can recognize that 0.986 is (1 - 1/71) nearly so take the total days equal to degrees then take back a degree every 71 days. 

    ‡ How can you check your value of LHA Aries? You can work it the long way, of course. You can also check using Stellarium. Go to the location and time of your scenario. Click on a star and look for Apparent Sidereal Time. This is identical to LHA Aries but given in units of time. If Stellarium shows SidT equal to 5:12:00, you can convert that to degrees the usual way: 15° per hour and 1° for every four minutes: 5:12:00 SidT is identical to 78° LHA Aries.

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