Sunday, August 14, 2016

Double Prime Meridian – why is the Greenwich Prime Meridian 102m west of the GPS zero meridian?

If you visit the Royal Greenwich Observatory and stand across the Prime Meridian of the World, your GPS will tell you that you are are at 0.00147°W (or 0°0.09′W or 0°0′5.3″W, depending on how your system displays coordinates), the GPS zero meridian (properly called the International Terrestrial Reference Frame zero meridian) is 102m further east. A recent article on, Moving the Earth's Prime Meridian, says that this is due to "improving technology" and "increased accuracy" in the measurements. This seemed unlikely, as the error sounds well above the level of accuracy achievable with the Airy Transit Circle, so I clicked through to the scientific article by Malys et al. that they were reporting on, and found that had completely misunderstood it. The actual reason is completely different, and far more interesting.*
* Well, to me anyway

GPS on the Greenwich Prime Meridian, showing 0.00149W
This GPS unit reads 0.00149°W on the Prime Meridian at Greenwich (Source: J. Cohen/Wikimedia Commons)
Airy defined the Prime Meridian using astronomical measurements of the stars. His instrument was surveyed carefully into position. He measured the horizontal with a bowl of mercury (don't do this at home, kids!), which was used to define the vertical. There was nothing wrong with his astronomical measurements – it was his measurement of the vertical that caused the position of the Prime Meridian to be 'wrong'.

What could possibly be wrong with that though? When we think of vertical we think of 'straight down' – the direction in which things fall, which is what Airy measured. However, this is not what is used for vertical when defining the Earth's coordinate system nowadays. For the coordinate system, a vertical line is one joining where we are on the Earth's surface to the centre of mass of the Earth. Globally this works, but locally on the Earth's surface there are deflections caused by mountains, ocean trenches, and other lumps and bumps. The mass of a mountain pulls the local vertical towards it, while the missing mass of a trench means the local vertical is pulled away from it.

Here at Arecibo Observatory we are just south of the Puerto Rico Trench – the deepest spot in the Atlantic. This causes a significant 'deflection of the vertical', as it is termed. The astronomical latitude of the Observatory, based on the vertical defined by the local gravity, is 18°21′13.7″N, but the geodetic latitude, based on the vertical passing through the Earth's centre of mass, is 18°20′36.6″N. That's a shift of 37.1 arcseconds, corresponding to over a kilometre! This is caused by the 'missing mass' of the Puerto Rico Trench, meaning the local 'straight down' is pulled south by the mass of Puerto Rico, shifting 'straight up' northwards. The stars that pass directly overhead are 37.1 arcseconds further north than our geodetic coordinates would indicate – we have to use astronomical coordinates, based on our local vertical, in order to point our telescope correctly.

The 102m shift at Greenwich is due to the deflection of the vertical in exactly the same way as the shift at Arecibo. Malys et al. used a model for the Earth's gravitational field to estimate the deflection of the vertical at Greenwich, and found that it was 5.5 seconds of longitude (and about 2.2 arcseconds in latitude, which isn't important here). This makes the astronomical longitude of the Greenwich Prime Meridian 0°00′00.19″ ± 0.47″. In other words, completely consistent with it being at zero longitude, as originally defined. This means that a line passing through the centre of mass of the Earth and the GPS zero meridian 102m east of Greenwich is parallel with a locally vertical line passing through the Greenwich Prime Meridian: they point to the same place in the sky. However, a locally vertical line on the GPS zero meridian would point about 5 seconds of longitude (3.3 arcseconds at that latitude) east on the sky, meaning a transit there would take place about 0.35 seconds earlier than predicted.
A line joining zero longitude on the celestial sphere to the centre of mass of the Earth passes through the GPS zero meridian, but zero longitude on the celestial sphere appears 'straight up' on the Greenwich Prime Meridian (which is how zero longitude on the celestial sphere was originally defined). As the celestial sphere is effectively at an infinite distance, the line from the Earth's centre of mass to zero on the celestial sphere and the line from Greenwich to zero on the celestial sphere are essentially parallel in reality, but that would require an infinitely large figure to show!
So, where does the "improving technology" come in? Satellite measurements force the vertical to be towards the Earth's centre of mass – which is what the satellites are orbiting. Furthermore, we now want a single system that will cover the entire globe – it is much easier to get a match between astronomical and geodetic coordinates when trying to only map a single continent. In North America, the ellipsoid (approximation to the shape of the Earth) worked out by Alexander Clarke in 1866 is generally off from the true shape of the Earth (i.e. sea level) by less than 10m, while the ellipsoid used for the GPS coordinate system is always off by at least 15m and often more. But the North American ellipsoid, and the 1901/1927 North American Datum coordinate system based on it, would fail dismally if extended over the entire world, while the GPS system works approximately equally badly everywhere.[1] This is why GPS heights are often significantly off from heights measured from sea level – the difference is about 60m in Arecibo.

In summary: The Greenwich Prime Meridian is the line of zero longitude defined astronomically, with vertical defined by local gravity. The GPS zero meridian is the line of zero longitude defined geodetically, with vertical passing through the centre of mass of the Earth. If you calculate a point on the sky that will be overhead at zero longitude at a certain time, it will be overhead on the Greenwich Prime Meridian, not the GPS zero meridian. Airy wasn't wrong, he was simply using a different definition of 'vertical'.

No comments:

Post a Comment