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    Re: Mars - Mercury Question
    From: Brad Morris
    Date: 2013 Feb 7, 13:17 -0500

    Hi Frank

    Yes, I think technically it would work.  Your result of 2.4 degrees per hour at transit is likely a closer than my estimate of 2.5 degrees per hour at transit, based upon the method of calculation.   As accurate as a lunar?  No!  Accurate enough to determine your longitude to a Prize value?  I think we agree, yes!

    Of course, Dr Maskelyne would have properly rejected me, strictly on the basis of utility.  13 or 14 times per century at 5 hours a time?  Woooohoo!  Thats 70 hours out of 100 years.  Not really useful at all.  I can just see my award from the Board of Longitude now, a really nicely worded letter, thanking me for my contribution.  And not a ha' penny more!

    Now for that once in 250000 Year event, when both the Transit of Mercury and Transit of Venus are simultaneously visible, please see http://www.fourmilab.ch/documents/canon_transits/  And the simulation (movie) of the event, please see http://www.fourmilab.ch/documents/canon_transits/movies/a26982532.gif

    "Yessir Dr Maskelyne, all I have to do is wait until the year 69163, and longitude is solved!"  Since the transits of both Venus and Mercury are simultaneously visible and moving at different rates, I should be able to determine my longitude, with more resolution than just the solitary transit.  It better not be cloudy that day!

    Regards
    Brad

    On Feb 7, 2013 12:28 PM, "Frank Reed" <FrankReed@historicalatlas.com> wrote:

    Brad,

    Good point about transits! Again, you can work out the speed with a really simple plane trigonometry diagram "looking down" on the Solar System. Or in the case of a transit, the triangle collapses, so consider this: Mercury is advancing at a Sun-centered rate of 4 degrees per day while the Earth is moving at a Sun-centered rate of 1 degree per day (we've known these heliocentric rates since the first heliocentric models of the Solar System). Relative to the position of the Earth in the sky as seen from the Sun, Mercury is advancing 3 degrees per day. Now let's flip it around and view this motion from the Earth. The Earth is further from Mercury than the Sun is. Since Mercury orbits at a distance of 0.4 AU and the Earth orbits at 1.0 AU, the Earth when Mercury is in transit is 0.6 AU from Mercury. So it's 1.5x further from Mercury than the Sun, and therefore the angular rate of Mercury across the sky relative to the Sun is two-thirds of the heliocentric rate or 2 degrees per day. There's some rounding down here, so let's call it 2.4 degrees per day. Note that the rate can be somewhat higher or lower depending on whether Mercury is near perihelion or aphelion since it has a relatively eccentric orbit, but that averages out so we can ignore it for now. That rate of 2.4 degrees per day is 0.1 degrees per hour. Divide both by sixty and we get 0.1 minutes of arc in one minute of time. Lunar distances change at a rate of about 0.1 minutes of arc in 12 seconds of time, so a rare Mercury transit is five times slower than a common lunar observation and therefore not as useful for determining longitude by a long shot. It would work, just not very well. Also, historically, the almanac data for Mercury was not particularly accurate until the early 19th century.

    There is, however, an important special case regarding Mercury transits and longitude determinations. You can do it without a high-quality sextant by timing contact events. So suppose you're an explorer in Australia in 1820. You have a decent telescope, and you have a rather low-grade octant and an artifical horizon, but you cannot afford a quality sextant. By good luck, the day of a Mercury transit arrives, and the sky is sparkling clear. You observe multiple altitudes of the Sun to get local time, before, during, and after the transit, and you look through the telescope and mark the instants when Mercury's tiny disk touches the limb of the Sun, perhaps counting off seconds to the next Sun altitude observation. Then you write up your observations and send them off to an astronomer with some skill who observed the same transit from a known longitude. Calculate... calculate... calculate. Wait for the mail. And then a letter arrives (on a Saturday!) informing you of your longitude based on your observations. Unfortunately, there was a typo in your first letter, and your longitude, as calculated, is in the middle of the South Atlantic. Bug detection is an iterative process. More letters are exchanged and eventually, weeks later, you know your longitude. Civilization has been advanced, and you move on to the next watering hole. :)

    -FER


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