GPS Satellites in View at High Latitude

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jimh
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GPS Satellites in View at High Latitude

Postby jimh » Tue Dec 05, 2017 4:14 pm

In the USAF NavStar global positioning system, commonly referred to as GPS, the satellites are in medium earth orbits with an inclination of 55-degrees. This means that the highest latitude the satellite orbit reaches is 55-degrees, North or South. If you are a user of GPS in high latitude, say at 55-degrees, you might think that all GPS satellites will appear to your point of view as being in the southern sky or at best overhead; I made that assumption myself, but discovered it was not correct.

If you are located at 55-degrees North latitude, then occasionally a GPS satellite should pass just about overhead, but never to your North--except that on the other side of the world those same satellite may also cross your meridian at 55-North latitude on that hemisphere. Using some simple geometry, we can see that a satellite that is on your meridian but anti-polar and reaches 55-N latitude will actually be visible to you with an elevation of roughly 6-degrees, and will appear to you to be due North. As a user moves to higher latitudes, more satellites over the North Pole will be visible to them low in their northern sky.

A recently published article in GPS World magazine included some azimuth/elevation plots of satellites in view and their tracks as seen from high latitudes in Finland. I reproduce one of them below to illustrate how GPS satellites from the other side of the earth show up in the northern sky of users in high latitudes:

SatellitesInViewFinland.jpg
Satellites In View from Northern Finland; adapted from
http://gpsworld.com/innovation-egnos-in-northeastern-europe/
SatellitesInViewFinland.jpg (20.33 KiB) Viewed 586 times


As can be seen in the polar plot of GPS satellite orbit tracks, many GPS satellites will be in view to the North to a user in high latitudes.

jimh
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Re: GPS Satellites in View at High Latitude

Postby jimh » Sun Dec 24, 2017 1:34 pm

There is a often made observation regarding GNSS satellites at high-latitude locations that "GLOSNASS is better than GPS." This notion comes from the difference in the INCLINATION of the orbital planes of the two systems:

GLONASS = orbit inclined 65-degrees
GPS = orbit inclined 55-degrees


We can compare the elevation angle from an observer in high latitude to a satellite in each system when that satellite is on the observer's meridian and reaches its highest elevation in the sky. A diagram below shows the relationship:

satellitDwg.jpg
Satellite elevation to observer at pole; drawn roughly to scale
satellitDwg.jpg (29.9 KiB) Viewed 533 times


In this diagram, we know the following dimensions:

b = Earth radius, 6371 km
c = Earth radius + satellite altitude; for GPS 26571 km; for GLONASS 25471 km
A = the difference between observer's latitude (90°) and satellite inclination; for GPS 35°; for GLONASS 25°


With two sides and one angle known, we can use the Rule of Cosines to solve for side a. Then we can find angle B from the Rule of Sines. And finally angle C from the definition of a triangle having 180-degrees of angles. (If you like, you can do this with the SIDE-ANGLE-SIDE calculator at https://www.triangle-calculator.com/?what=sas. Just enter the three values (a, b, C) in the calculator fields.) We find that:

For GPS, C = 135.3°
For GLONASS, C = 147.2°


By subtracting 90 from C, we get the elevation angle to the satellite from the observer to the satellite:

For GPS elevation = 45.3°
For GLONASS elevation = 57.2°


We find that for an observer at the Pole, a GLONASS satellite can rise as high as 57.2° in the sky, while a GPS satellite can only rise to 45.3°. This gives the GLONASS satellite an 11.9° advantage at the Pole.

jimh
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Location: Michigan, Lower Peninsula
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Re: GPS Satellites in View at High Latitude

Postby jimh » Tue Dec 26, 2017 10:12 am

Going back to the observer at 55-degree-N latitude: a GPS satellite on his meridian over the pole could also reach a maximum latitude of 55-degrees. Since the observer and the anti-polar satellite are both 55-degrees above the equator, the angle between the observer and the satellite must be 70-degrees (from 180 - 55 -55). A sketch shows the arrangement:

antiPolarGPSinView55N.jpg
GPS satellite on observer's meridian over the pole
antiPolarGPSinView55N.jpg (37.2 KiB) Viewed 506 times


The angle B is shown in detail at the lower right. At the observer's location, his horizon is at 90-degrees to side a. To find elevation we subtract 90 from angle B; this gives an elevation angle of 6.21-degrees.

As the observer moves farther north, the satellite elevation increases.

(I include just a sketch; it takes me way too much time to make an decent digital graphic compared to a few moments to draw a sketch and scan it.)