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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Refraction -- A general formula for all heights and altitudes**

**From:**Frank Reed CT

**Date:**2005 Aug 25, 14:59 EDT

For the calculating members of the list, here's a general fast formula for 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. -FER 42.0N 87.7W, or 41.4N 72.1W. www.HistoricalAtlas.com/lunars