### The Satellite Clock Bias, dt

One of the largest errors can be attributed to the satellite clock bias. It can be quite large, especially if the broadcast clock correction is not used by the receiver to bring the time signal acquired from a satellite’s on-board clock in line with GPS time. As time is a critical component in the functioning of GPS, it is important to look closely at the principles behind this bias.

The onboard satellite clocks are independent of one another. The rates of these rubidium and cesium oscillators are more stable if they are not disturbed by frequent tweaking and adjustment is kept to a minimum. While GPS time itself is designed to be kept within one microsecond, 1 µsec or one-millionth of a second, of UTC, excepting leap seconds, the satellite clocks can be allowed to drift up to a millisecond, 1 msec or one-thousandth of a second, from GPS time.

One of the largest errors is the satellite clock bias. There is more than one clock in each GPS satellite. There are cesium and rubidium clocks (frequency standards) and they are quite stable. Nevertheless, one of the largest biases is attributable to these clocks.

The broadcast clock correction in the Navigation Message has been mentioned earlier. This is the correction that the control segment provides to the receiver to bring the satellite clock in line with GPS time. The clocks are kept within a microsecond —that's a millionth of a second— of UTC, excluding leap seconds, of course - but still, the satellite clocks are allowed to drift up to a millisecond —or thousandth of a second— from GPS time.

The control segment doesn't want to constantly tweak the clocks, as this would cause the clocks to deteriorate more rapidly. I think it's fair to say that the clocks are one of the weakest aspects of the satellites, although the GPS satellites have done very well. Frequently, what causes them to break down are the clocks. Since constant tweaking would diminish their longevity, they are allowed to drift up to a thousandth of a second from GPS time.

### Relativistic Effects on the Satellite Clock

Albert Einstein’s special and general theories of relativity apply to the clocks involved here. At 3.874 kilometers per second, the clocks in the GPS satellites are traveling at great speed, and that makes the clocks on the satellites appear to run slower than the clocks on earth by about 7 microseconds a day. However, this apparent slowing of the clocks in orbit is counteracted by the weaker gravity around them. The weakness of the gravity makes the clocks in the satellites appear to run faster than the clocks on earth by about 45 microseconds a day. Therefore, on balance, the clocks in the GPS satellites in space appear to run faster by about 38 microseconds a day than the clocks in GPS receivers on earth. So, to ensure the clocks in the satellites will actually produce the correct fundamental frequency of 10.23 MHz in space, their frequencies are set to 10.22999999543 MHz before they are launched into space.

There is yet another consideration, the eccentricity of the orbit of GPS satellites. With an eccentricity of 0.02, this effect on the clocks can be as much as 45.8 nanoseconds. Fortunately, the offset is eliminated by a calculation in the GPS receiver itself; thereby avoiding what could be ranging errors of about 14 meters. The receiver is moving, too, of course, so an account must be made for the motion of the receiver due to the rotation of the earth during the time it takes the satellite's signal to reach it. This is known as the Sagnac effect, and it is 133 nanoseconds at its maximum. Luckily, these relativistic effects can be accurately computed and removed from the system.

The satellite clocks are subject to relativistic effects as described by Einstein. They are traveling at a great speed in orbit, and they are in weaker gravity. This means that the GPS satellite clocks do appear to run a slight bit faster than the clocks in the GPS receivers.

The fundamental frequency for the GPS system is 10.23 megahertz. The GPS satellite clocks must appear to an observer to be running at that frequency. Therefore, before they are launched, their frequency is actually set just a slight bit slow, and they appear to actually run at 10.23 megahertz to an observer, because Einstein was correct.

### Satellite Clock Drift

The onboard satellite clocks are independent of one another. While GPS time, itself, is designed to be kept within one microsecond, 1 µsec or one-millionth of a second, of UTC, excepting leap seconds, the satellite clocks can be allowed to drift up to a millisecond, 1 msec or one-thousandth of a second, from GPS time. There are three kinds of time are involved here. The first is UTC per the U.S. Naval Observatory (USNO). The second is GPS time. The third is the time determined by each independent GPS satellite.

Their relationship is as follows. The Master Control Station (MCS) at Schriever (formerly Falcon) Air Force Base near Colorado Springs, Colorado gathers the GPS satellite’s data from monitoring stations around the world. After processing, this information is uploaded back to each satellite to become the broadcast ephemeris, broadcast clock correction and so forth. The actual specification for GPS time demands that its rate be within one microsecond of UTC as determined by USNO, without consideration of leap seconds. Leap seconds are used to keep UTC correlated with the actual rotation of the earth, but they are ignored in GPS time. In GPS time, it is as if no leap seconds have occurred at all in UTC since 24:00:00, January 5, 1980. In practice, the rate of GPS time is much closer than one microsecond of the rate of UTC; it is usually within about 25 nanoseconds, 25 nsec or 25 billionths of a second, of the rate of UTC.

By constantly monitoring the satellites' clock error, dt, the Control Segment gathers data for its uploads of the broadcast clock corrections. You will recall that clock corrections are part of the Navigation message.

So, there is a difference between GPS time and UTC. UTC and GPS time were exactly the same at hour 24 on January 5, 1980. From that moment, GPS time continued on, but without leap seconds. The broadcast clock correction from the satellites allows the receiver to get close to GPS time.