Authors' Mistakes #17 - Peter David

As far as I know, Peter David has written 95 novels, mostly SF, and won 10 awards. I know him from his Star Trek books, of which, between 1997 and 2002, I read 19. I am now reading After Earth, the novelisation of the recent film with Will Smith, and discovered in it an appalling (and certainly unexpected) mistake.

On page 101, he wrote: A parsec, she recalled, was a measure for the speed of light, how far it would travel over one hundred years.

The sentence is at best awkward. What does it mean “A parsec [...] was a measure for the speed of light” when in fact a parsec is a measure of distance, as David writes in the next sentence?

But the problem is that a parsec is only 3.26 light years, not 100!

If you draw two lines from a point in interstellar space, one passing through the sun and one passing through Earth, when the amplitude of the angle between the two lines is one second of arc, the object’s distance is by definition 1 parsec. The “par” in parsec stands for “parallax” and the “sec” for “second”.

Imagine making two observations of a star with a six months period between them. During the six months, Earth will have moved half of its orbit. As a result, you will have to point the telescope in two slightly different directions. If you know the radius of Earth’s orbit, you can use the angle between the two directions to calculate the star’s distance.

Here is how you do it.

Earth’s distance from the sun (i.e., the radius of Earth’s orbit) is approximately 150 million km = 1.5 x 10⁸ km.

A second of arc is 1/3600 of a degree and there are 360 degrees in a full circle, which is 2π times the radius R of the circle. This means that 2π x R / 360 / 3600 is the length of a second of arc = R x 4.85 x 10^-6.

If a star has a parallax angle of two seconds (not one second because the two lines of view are through opposite points of Earth’s orbit, rather than one though Earth and one through the sun), to calculate its distance in kilometres you only need to imagine a circle of radius D centred on the star and passing through the sun. Then, the diameter of Earth’s orbit is given by:

1.5 x 10⁸ km x 2 = 2 x D x 4.85 x 10^-6

That distance will be 1.5 x 10⁸ km / 4.85 x 10^-6 = ~3.1 x 10¹³ km.

As the speed of light is 300,000 km/s = 3 x 10⁵ km/s, 1 light year is 3 x 10⁵ km/s x 3600 s/h x 24 h/d x 365 d/y = ~ 9.46 x 10¹² km (actually, the light is 0.07% slower, but the year is 0.07% longer, so it works out just fine! :-) .

Then, 1 pc = ~3.1 x 10¹³ km / (9.46 x 10¹² km/ly) = 3.28 ly. Close enough, considering the approximations.

There is no star at 1 pc from Earth, but Proxima Centauri, the closest star to Earth, is 1.3 pc away.