# Unfortunate Satellite Launch Problem Allows Test of Relativistic Time Dilation

Quote of the Day

Happy families are all alike; every unhappy family is unhappy in its own way.

— Leo Tolstoy from 'Anna Karenina'. This quote is the basis of the Anna Karenina Principle, which describes an endeavor in which a deficiency in any one of a number of factors dooms it to failure. Consequently, a successful endeavor (subject to this principle) is one where every possible deficiency has been avoided.

## Introduction

Figure 1: Galileo Navigation Satellite (Source).

In November 2015, the European Space Agency (ESA) had a launch problem with two of its Galileo navigational satellites that resulted in both satellites being placed into highly elliptical orbits. ESA can burn some of the satellites' station-keeping fuel to bring these orbits back to standard, but this will take some time. While the orbit adjustments are occurring, ESA will use the satellites to provide another test of Einstein's general theory of relativity. Specifically, they will test the prediction that clocks will run slower the closer they approach a massive object.

I read about this experiment in an article in The Register. Normally, I would not comment on an article like this one; however, this article had an error in it that bothered me. The article states that an earlier test of general relativity measured a rather large clock rate increase:

Gravity Probe A found that a clock 10,000 kilometers up ran 140 parts in a million of a second faster than the same device on Earth, but it was a one-shot mission.

This number could not possibly be correct. I deal with precision clock sources every day – typically Stratum 1 level, and a clock variation of 140 ppm would be easily visible using standard watches. If my watch were running 140 ppm fast, that would mean that it would run 12 seconds fast every day – I would notice an effect this large very quickly. The effects of general relativity are very real, but are much more subtle. I thought I would determine what the article should have reported.

I believe what the article's author intended to say was that the Gravity Probe A experiment verified general relativity to a precision of ±70 ppm, which is what the project quoted in its original report.

## Background

### Gravity Probe A Time Dilation Result

I found a reference to the time dilation measured by Gravity Probe A in the book Splitting The Second: The Story of Atomic Time (ISBN 9781420033496). Here is a quote from this book.

But the most spectacular conﬁnnation of the gravitational shift came in June 1976, when a NASA spacecraft called Gravity Probe A was launched on a rocket to a height of I0 000 kilometres before falling back to Earth. The probe carried a hydrogen maser clock constructed by physicists Roben Vessot and Martin Levine at the Smithsonian Astrophysical Observatory. As usual, two relativistic effects were operating: time dilation due to the speed of the rocket and the gravitational shift due to the height above sea level. By monitoring the speed of the rocket throughout the 2-hour ﬂight. the physicists were able to separate out the two effects and show that at the maximum height the gravitational shift was causing the clock to run fast by four parts in 1010. as predicted by general relativity. The agreement was within 70 parts in a million.

If you want to read more about time dilation, I have included a reference to this book in this post's Appendix A.

### My Previous Writing on This Subject

I went into detail on the clock shifts associated with GPS satellites in this post. The same math is used here.

## Analysis

The calculation of the time dilation due to a satellites position in a gravity field is straightforward and shown in Figure 2. I show a time dilation of about 4 parts per 1010, as expected.

Figure 2: Quick Calculation of the Time Dilation due to General Relativity.

## Conclusion

I find errors in newspaper and magazine articles all the time. Normally, I just ignore them. However, this one bugged me because it seemed so far off, and it is on a topic that I work with every day.

I should comment that some publications are excellent at making corrections. I found an error in New York Times article a few years ago and sent a note to the author. He corrected the online version within a few hours.

## Appendix A: Test Reference

This excerpt is from the book Splitting The Second: The Story of Atomic Time, which I quoted in the post.