How does your GPS device know where you are?
Since the 1980s, the US government has made
the radio signals from their network of GPS
satellites, orbiting over 20,000 km above
the Earth’s surface, available to anyone
with an appropriate GPS receiver. Worldwide
coverage by 24 satellites arrived in 1993,
and it became really useful for a whole range
of civilian purposes when access was fully
opened up in 2000.
More recently, other services have been launched
by Russia, China, India, Japan, and the European
Union, to run either independently, or to
enhance the accuracy of the American system.
The satellites transmit geolocation and time
information, along with a generated stream
of pseudocode, and by comparing signals from
four or more satellites, a GPS receiver can
reliably calculate where in the world it is
within a few metres. With more satellites,
and information from additional ground-based
systems, the location can be calculated within
cm.
In this video we’ll be taking a look at
how satellite navigation works, some of the
calculations involved, and how we need to
use Einstein’s special and general relativity
to make it as accurate as it is!
First though, let’s imagine we’re lost
in a featureless two-dimensional flat landscape
who’s only redeeming feature is that it
exists within a clearly defined coordinate
system with units measured in km east of a
known location (the x coordinate), and km
north of that location (the y coordinate).
We’re pretty sad about being lost.
Then suddenly, a magical being appears, and
we ask them where we are. They tell us that
we’re exactly 39 km away from the centre
of Antville, which has coordinates (14, 45).
We quickly whip out some graph paper, make
a grid, and draw the circle on it.
So, we could be anywhere on the circumference
of a circle with a radius of 39 km, and its
centre in Antville.
We can even write down the equation of the
circle of points where we could be.
(x-14)^2 + (y-45)^2 = 39^2
Now, that’s helpful-ish, but there are still
infinitely many places we could be!
OK, so I’m being a bit pedantic to say that
there are infinitely many positions around
that circumference, but there ARE!
We’ve certainly narrowed down our position
in the world quite significantly, but we don’t
know exactly where we are, and we’re probably
wondering why the magical being didn’t just
tell us our own coordinates rather than giving
us the cryptic reference to Antville.
Perhaps they were a maths teacher in a previous
life?
Then, as if reading our thoughts, the magical
being tells us that we are also exactly 50
km away from the centre of Buffalo Town, which
has coordinates (80, 70). Now that is quite
helpful.
We add that circle to our graph paper, and
write down its equation: (x-80)^2 + (y-70)^2
= 50^2.
If Antville and Buffalo Town were exactly
39 + 50, that’s 89 km apart, then these
two circles would touch in just one place,
and we’d now know exactly where we were.
Sadly, they’re not, and there are two places
where the circles intersect, and we could
be in either one of those.
Then the magical being tells us that we are
exactly 29 km from the centre of Cat City,
which has the coordinates (71, 50).
We draw out that circle, and write down its
equation (x-71)^2 + (y-50)^2 = 29^2, and discover
that it cuts each of the previous two circles
in two places, but all three circles only
intersect at one point, with coordinates (50,30).
Now, if all of the information is 100% accurate,
and the magical being was telling us the truth,
then we now know exactly where we are.
If the centres of the circles had all fallen
in a straight line, then they’d probably
all intersect in two places, so we wouldn’t
be able to narrow down our position to just
one place, but if we can cut a deal with the
magical being to make sure the information
they give us isn’t from circles with their
centres all in a straight line, then we can
work out exactly where we are!
Now, we used scale drawing to work out our
position, but we could also use a bit of algebra
– solving the simultaneous equations – to
work it out more accurately. The other advantage
of the algebra, is that we could program a
computer to work out the answer for us very
quickly!
This is a kind of simple two-dimensional analogy
to satellite navigation.
When we extend the coordinate system to three
dimensions, we get spheres of possible positions,
rather than circles, and we need four spheres
with their centres not in a straight line
to work out the x, y, and z coordinates of
our location, rather than three, but it’s
still just a matter of solving simultaneous
equations.
This technique of using circles or, in the
case of GPS, spheres, from known reference
points to work out a location is known as
trilateration.
But it’s just part of the process that your
GPS receiver device uses to work out where
you are.
A real-world communication system doesn’t
have a magic being that can tell you the exact
coordinates and distances to known locations!
Instead, in our 2-D analogy, let’s say that
we have radio transmitters at the centres
of Antville, Buffalo Town, and Cat City, and
that they are transmitting time-coded messages
with their coordinates.
The radio transmitters know the time super-accurately,
because they have atomic clocks, and they
also know exactly where they are.
Radio waves are electromagnetic waves that
travel at the speed of light, 299 792.458
km per second. Well, that’s the speed in
a vacuum, and it’ll vary slightly through
the atmosphere, but let’s not worry about
that yet.
To make things simpler in our 2-d example,
let’s say we’ve got special radio waves
that travel at only 1 km per second.
So, then, we’re standing in the featureless
landscape, not knowing where we are, we’ve
got our receiver, and the signal takes 39
seconds to get from Antville, 50 seconds to
get from Buffalo Town, and 29 seconds to get
from Cat City. If the clock on our receiver
is perfectly synchronised with the clocks
in the transmitters, and continues to run
at that same rate, then we can examine the
timestamps in the messages, and compare them
with the time when they arrived. And then,
we can work out how long it took them to get
here.
We know that speed = distance / time, so we
can rearrange this into distance = speed times
time. The signal from Antville, for example,
took 39 seconds to get here, and the speed
of the signal is 1 km per second, so the distance
to Antville is 1 km/s times 39 s, that’s
39 km.
Similarly, we can work out the distances to
Buffalo Town and Cat City from the difference
in the timestamp when the messages were sent,
and when we received them.
Since the messages also contain the coordinates
of the towns, it’s as if our magical being
is back, giving us the information, and we
know we can work out exactly where we are.
So, back in the 3-dimensional real world,
GPS satellites do all have super-accurate
atomic clocks on-board, and are programmed
to simultaneously transmit their coordinates
in three-dimensional space, along with the
corresponding accurate timestamp.
However, the average GPS receiver doesn’t
have such an accurate clock – they'd be
far too big and expensive to put in your smartphone.
This means that the time delay between the
signal setting off, and arriving at your receiver
can’t be measured as accurately as we’d
like!
If the clocks were badly de-synchronised,
then the calculations would tell you that
you were in quite a different place to where
you actually were!
Luckily, most receivers cleverly repeatedly
analyse the messages from lots of satellites,
and reverse-engineer the calculations we did
earlier to compare where they actually are
with where the time difference implies they
are.
This enables them to update their internal
clocks to keep much more accurate time. GPS
receivers don’t need atomic clocks, because
they can effectively piggy-back off the atomic
clocks in the satellites instead. The more
satellites your GPS receivers can see, the
more data they have to work with, and the
more accurate they can be.
In 3 dimensions, you generally need at least
four signals from separate satellites to work
out where you are, although if you have three
signals it may be possible to use the centre
of earth as a kind of fourth satellite, assuming
you’re on the earth’s surface, but this
will almost certainly give you a less accurate
idea of where you are.
If you are within sight of 7 or more satellites,
and conditions are ideal, it’s possible
to get results within just a few metres of
your true position around 95% of the time
with publicly available GPS systems, and this
can be improved further with additional information
from supplementary ground-based systems.
Even with the clever clock updates and multiple
satellite readings, radio signals can bounce
off of buildings, trees, and even clouds,
and will slow down slightly in denser parts
of the atmosphere, which can affect how long
they take to reach your receiver. This makes
it very tricky to know when your location
reading is highly accurate, and when it’s
less accurate.
But, there’s one even more surprising thing
that satellite navigation systems have to
take into account when working out where you
are.
RELATIVITY!
Because the satellites are travelling so fast
in orbit around the earth, we need to use
this time dilation formula when evaluating
time from the point of view of the clock on
the satellite.
In our case:
Delta t is the time in seconds as observed
by a theoretical observer on Earth
Delta t sub zero is the time in seconds that
registers on the atomic clock on board the
satellite
V is the velocity of the satellite in metres
per second
C is the speed of light in metres per second
Now, there are 86,400 seconds in a day, and
the speed of light is 2.998 * 10^8 m/s, but
what is the velocity of the satellite?
Well, GPS satellites orbit 20,000 km above
the surface of the earth, and the earth has
a radius of about 6,371 km. So the circumference
of the orbit, assuming it’s a circle, is
2 times pi times the radius of the orbit … and
that makes 165,693,879.7 metres.
It takes the satellite 11 hours and 58 minutes
to orbit the earth once – that's 43,080
sec.
This means the satellite travels at about
3,846.19 m/s, or 13,846 km/h, which is quite
fast.
When we plug those values into the formula,
we see that 1 day seems like a day plus 7
microseconds to the satellite! It’s going
to lose 7 microseconds per day compared to
a clock on the Earth, even with its atomic
clock.
But that’s not all. The satellites are over
four times further away from the earth’s
centre of mass than we are on the ground,
which makes the gravitational effect of all
that mass weaker.
In fact, the gravitational effect varies as
the inverse square of the distance from the
mass, so gravity seems over 17 times stronger
to us on the ground than it does to the satellite.
That means that spacetime is warped less up
there, and clocks run faster than they would
here on Earth.
We can express the differences in timing observations
made on Earth compared to the satellite due
to gravity using this formula.
Where:
Delta t prime is the time in seconds that
passes on the observer-under-the-influence-of
gravity’s clock
Delta t is the time in seconds that registers
on a theoretical observer-not-affected-by-gravity’s
clock
G is the universal gravitational constant
M is the mass of the gravitational object
(Earth in our case) in kg
R is the distance in metres from the gravitational
object
And c is the speed of light in metres per
second
The clocks run 45 microseconds a day faster
on the satellite than they would here on Earth.
So, 7 microseconds slower per day due to high
velocity, and 45 microseconds faster a day
due to lower gravity. That makes an overall
difference of 38 microseconds a day.
If we didn’t take that into account, the
timings would drift by 38 microseconds a day,
and since the radio signals travel at the
speed of light, that would mean our measurements
would drift out by about 11.4 km per day – satnav
would be useless.
You could say that we wouldn’t be where
we are today without Einstein’s special
and general theories
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