Celestial Navigation Basics – Hc Calculated Altitude

In this post we will look at how to determine Hc the calculated altitude and compare this to Ho the observed altitude. The difference between Ho and Hc is the intercept and is used in the Marc Saint Hilaire method for solving position. In previous posts we found Hs (Ref.3), Ha (Ref.4) and Ho (Ref.5). The complete details are contained in Ref.1. Complete Sight Reduction procedures are contained in the Nautical Almanac Ref.2 starting page 277, and the NAO form on page 319.

Hs = Sextant Altitude

Ha = Apparent Altitude

Ho = Observed Altitude

Fig.1 shows the sight reduction of our sextant altitudes taken on Monday July 6th_2020 at Trillium Park (Ref.3):

Lat = 43deg 37′ 46.2″ North
Long = 79deg 24′ 34.2″ West
Height of Eye = 3.0m
Index Error = 59min = 1min on_limb

Hs average = 63deg 32.25min at 12:06DST/16:06UTC

Fig.1 Sight Reduction Hs, Ha, Ho, Hc, Intercept and Zn.

The first image under the blog post title shows the geometry involved in the calculation of Hc. It shows a plane intersecting along the great circle distance between the Observer C and the Sun GP A and going through the Earth centre. The Geographical Position or GP is the point where a line between the Sun centre and the Earth centre intersects the surface of the Earth. The Nautical Almanac lists the GP for the Sun, Moon, planets and 57 navigational stars by hour for a whole year.

The great circle distance between the Observer C and the SunGP A subtends an angle ZD (Zenith Distance) at the Earth centre. This angle ZD is equal to 90deg – Hc. So knowing ZD, gives us Hc. Fig.3 shows the Navigational triangle formed by the North Pole, Observer Location C and SunGP A. Fig.2 shows a ScicosLab script to solve this spherical triangle using the Spherical Cosine Law (we know sides a,c and contained angle B). Hc = 63.690deg = 63deg 41min. Azimuth = 136.9deg.

Hc can also be calculated using the Nautical Almanac NAO Concise Sight Reduction Form. In order to keep the tables manageable, the assumed position in Latitude is taken to be a whole number of degrees in this case 44degW. Also the assumed Longitude is adjusted so that the LHA is a whole number of degrees as well, in this case 79deg 16.6minW. Hc = 63deg 28min, Z =136.8deg. Note that this is for a slightly different Observer location, so naturally Hc and Z are also slightly different.

Fig.2 Great Circle Distance, Bearings and Hc between Toronto HF1 & SunGP
Fig.3 Navigation Triangle North Pole – Toronto HF1 – SunGP
Fig.4 Nautical Almanac Concise Sight Reduction Form Lat=44deg Long=79deg16.6min
Fig.5 Solution Spreadsheet NavPac Check/Google Earth Check/NAO Form
Fig.6 YouTube Video “Celestial Navigation Basics – Hc Calculated Altitude”

Download “Celestial Navigation Basics & Equipment”


#1. “Celestial Navigation Basics & Equipment”, Clark Telecommunications, 2019, ISBN 9780988049086

#2. – “Nautical Almanac 2020 Commercial Edition”, Paradise Cay Publications, ISBN 9781951116033

#3. – “Celestial Navigation Basics – Hs Sextant Altitude”

#4. – “Celestial Navigation Basics – Ha Apparent Altitude”

#5. – “Celestial Navigation Basics – Ho Observed Altitude”

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By Jeremy Clark

Jeremy Clark is a Senior Telecommunications Engineer and Advanced Amateur Radio Operator VE3PKC. He is the author of E-Books on Telecommunications, Navigation & Electronics.