NavList:

That is an amazing discovery that you made, that you can use the distance to the horizon table from Dresionstok ( and also other books have the same table) to do the computations for a visual observation of an object on the visible horizon and this observation can be made without any instruments. The reason that I used the word "discovery" is that there is no mention of this use accompanying the tables or in any navigation manual or textbook. The normal way to do this computation is to use the archaic "time sight" formula in use prior to the development of the LOP method of navigation. This formula is: cos LHA = ((sin H - (sin lat sin dec))/(cos lat cos dec)) To use this standard formula you input the altitude you are interested in and you get LHA as the output. Then to determine the time of this event subtract the LHA from the longitude to determine GHA of the body and then consult the almanac to see what time the body has this GHA. Bowditch say to use this formula if you want to determine the time of sunrise and you then use the altitude of the center of the sun, Modern Bowidtch says to use 16' for the SD and 34' for the refraction plus the dip for your height of eye. The 1914 edition of Bowditch say to use 36.5' for the refraction. To use this formula from an airplane you need to know the dip and the refraction from altitude. From 10,000 feet the dip is 97' refraction is 50' and SD is 16' making the center of the sun  -  2° 43' when you observe the upper limb and this is the value you put into the formula to determine the time of sunrise over Howland at 10,000 feet, I got 17:37:11, exactly the same as your computation. I was really surprised that your method came up with the same answer. Your formula produces the total of the refraction and the dip from altitude or this wouldn't work and that was the surprise. The formula for the dip is .97' times the square root of the altitude in feet. The formula for the distance to the horizon is in the same form, 1.15 times the square root of the height of eye in feet. This means that the formula  0.18 times the square root of the height of eye in feet plus 34 produces the refraction of an horizon observation as measured from altitude, and , amazingly, this simple formula produces a very good approximation of the refraction. The reason that I find this to be amazing is that the standard formulas for refraction are much more complex. We have had long discussion  on NAVLIST about the most accurate formulas to use for computing refraction. The links below will take you to some of those discussions. I have included the formula for computing refraction for negative altitudes like those measured to the horizon from an airplane, you can see it is much more complex. --- I have attached a copy of a piece of paper that has been my wallet for 35 years that I made by combining the refraction from the Air Almanac which provides for negative altitudes, the dip and, for the sun, the SD for both upper and lower limb shots. I used this mainly for amusement when I happened to be in flight when the sun was rising or setting and then I used the Ho from this table to calculate a standard LOP, not particularly accurate but still fun to do. gl http://fer3.com/arc/m2.aspx/Refraction-general-formula-for-all-heights-altitudes-FrankReed-aug-2005-w25346For the calculating members of the list, here's a general fast formula for http://fer3.com/arc/m2.aspx/Refraction-general-formula-for-all-heights-altitudes-FrankReed-aug-2005-w25348 refraction covering all observer heights and angles below the horizon, too: >>>>> kk=180/pi 'Alt is the angular height (corrected for dip). Ht is the observer's height above sea level. IF Alt < 0 THEN 'use new below zero degrees altitude formula: refx = EXP(3.537 - .369 * Alt + .051 * Alt * Alt) ELSEIF Alt < 15 THEN 'use standard low altitude refraction formula (Bennett): refx = .998 / TAN((Alt + 7.31 / (Alt + 4.4)) / kk) ELSE 'use standard high altitude refraction: refx = .972 / TAN(Alt / kk) END IF 'refx as calculated so far is the sea level refraction at standard temperature and 'pressure of 10 deg C and 1010 millibars. The result is in minutes of arc. 'Now adjust for non-standard tempreature and pressure. 'Temp and Press are sea level temperature in deg C and pressure in mbar: refx = refx * (Press / 1010) * (283.15 / (273.15 + Temp)) 'Now scale for height above sea level: 'Primary scaling with height above sea level: refx = refx * EXP(-ht / (11278 - ht / 13)) 'A correction in scaling for very low angular altitudes, proportional to observer altitude: refx = refx - (ht / 10000) * EXP(-Alt / 14) 'refx is in minutes of arc. <<<<< This procedure calculates refraction at all altitudes from sea level to at least 8km at excellent accuracy levels. It is based on (time-consuming) integrations using the method outlined in Auer-Standish with an atmospheric model compatible with the refraction tables in the Nautical Almanac (pre-2004), specifically a lapse rate of 7.25 deg C up to 11km altitude and constant temperature above that altitude. The fit to the integrations is accurate to 0.15 arcminutes or less except in cases where the refraction is very large in which case the angular error may be a few tenths of a minute of arc but is less than 1% of the total refraction. Since red light and blue light from a star have refractions that differ by more than 1%, this error limit should be acceptable in almost every case. This calculation will typically be thousands of times faster than running a complete integration. If computing time is not an issue, the integration approach is better. I neglected to mention that Alt is input in degrees and Ht is in meters. Alt should be between -3 and 90. Ht should be between 0 and 8000m -- outside this range is ok but with lower accuracy.  -FER 42.0N 87.7W, or 41.4N 72.1W. www.HistoricalAtlas.com/lunars http://www.fer3.com/arc/sort2.aspx?y=200508su&y2=200508&author=&subject=