A matter of small and big numbers

Having mentioned the difficulty of finding the distances to stars last week, our astronomer tries to put some unfamiliar numbers within your grasp.

Our astronomer writes:

Last week I mentioned that the distance to a star was a difficult thing to determine, and in the old days astronomers were forced to take short cuts.  I’d like to clarify just how difficult it was (and to some degree still is).

First, we need to set a scale.  We’ll be measuring small angles, expressing them in seconds of arc.  If I say there are sixty seconds in a minute of arc and sixty minutes in a degree, that allows you to do math but is hard to grasp.  If I say a second of arc is the size of a dime as seen from two miles away it’s not much better; I don’t know anyone who looks at pocket change from the next town.  Instead I’ll say that a second of arc is about how big a star appears in a small telescope under a very steady atmosphere.  (In an unsteady atmosphere it appears sort of like looking through a swimming pool at a light on the bottom, when people are splashing around in it: a large, dancing blob.  The atmosphere is unsteady most of the time.)

Now we go back to the idea that the Earth is not fixed at the center of the universe, but orbits the Sun.  One objection to this idea was that, if the Earth moved, the stars should appear to shift direction (“parallax”).  No one had been able to measure such a shift even with the most careful of naked-eye measurements, which are good to maybe a tenth of a degree (six arc-minutes).  That meant the stars were more than 573 times the radius of the Earth’s orbit away, which seemed absurdly large (or else the heliocentric theory was wrong).

When telescopes came into use smaller angles could be measured, and some changes in direction were detected.  But they didn’t behave in the right way!  It turns out that Earth acts like a spinning top, slowly shifting its axis, by about 50 arc-seconds per year.  That was known to the ancients.  But on top of this precession is a wobble called nutation, happening much faster, of 1.2 seconds superimposed on another of nine seconds.  Since all our angles are eventually referred to the Earth, this motion has to be carefully subtracted from our readings.

There was a shift in the measured direction to the stars, a sizeable one: 20.5 seconds.  And it varied annually, as one would expect.  But it was the same for every star in a given direction, and again didn’t match what would happen by shuffling back and forth in the Earth’s orbit.  This “aberration” was finally explained as the effect of the speed of the Earth compared with that of light!  As the telescope is carried along at 30 kilometers per second, it has to tilt at this tiny angle to catch the light coming in at 300,000 kilometers per second.  One could get rid of precession, nutation and aberration by measuring the position of a star with respect to a much more distant one right next to it in the sky; but how do you choose one that’s more distant?  We’ve already realized that fainter does not necessarily mean farther.  And most stars are in fact members of double-star systems, which would show exactly the same parallax and thus none compared to each other.

On top of these was a straight-line motion for many stars, amounting to a few seconds of arc per year.  But this “proper motion” gave a clue to stellar distances: those with large motions are, on the average, closer to us.  This assumption does not depend on stars being the same or similar brightness, as our earlier assumption did.  It only requires that we not be in a special place in the universe.  So astronomers concentrated their attention on these high proper-motion stars.

The first one accurately measured (there were several false alarms) was 61 Cygni, barely visible to the naked eye, with a parallax of 0.3 seconds.  This is after precession, nutation and aberration, all much larger effects, were accounted for.  It’s about the radius of the star as seen in the largest telescope in the world at the time (1838) under good conditions.  It was a great accomplishment, done not by inspiration or genius but by careful hard work.