The sweep of the night sky is as awe inspiring today as it always has been. In times past our ancestors supposed the eternal stars to be campfires of all powerful beings or immortal gods impervious to change and thus free from imperfection and decay. But whatever our early musings, deciphering the heavens has been slow and painful. Only over many centuries have we painstakingly been able to map the sundry motions and chart the depths. Indeed, we only discovered galaxies, and thus know the diameter of the universe to be about 90 billion light years, in the last century and just barely within the bounds of living human memory.
Today we have both direct and indirect methods of measuring stellar distances. In the less accurate indirect method, we estimate how intrinsically bright a star is. This “luminosity” or total amount of light emitted depends only on the temperature, or color, and the width of the star. Knowing this solves the problem of whether a particular star is a bright one far away or a dim one closer. In detail, the brightness decreases as the inverse square of its distance from earth.
For nearby stars, we have a direct and much more accurate technique called “parallax” which depends only on simple geometry. And knowing a few unmistakable distances via this method allows us to develop stellar models based on spectra and to group these closer stars into different categories. Each category has roughly the same intrinsic brightness creating what astronomers call “standard candles”. Assuming stars much further away have the same properties as those nearby, this gives us a yardstick for the cosmos.
Parallax is our most important and trustworthy technique. It relies on the fact that as we move, nearby objects appear to shift their positions more than objects which are further away. The standard demonstration is to ask you to stretch out your arm and hold up your thumb. Then close one eye only and then the other. Your thumb will appear to move slightly with respect to the distant background.
The same thing happens for stars as the earth swings around the sun. Most stars are so far away that despite heroic efforts, we can see no shift whatever against the distant background. These fixed stars form a dense matrix of well known constellations. But a relatively few do shift their position ever so slightly and by amounts we can just barely measure. Basically the target star appears to move in an ellipse against the distant background . The trick is to measure the angle, θ, between that star and the center of the ellipse. This is sometimes made more difficult if the star is not visible in the night sky during part of the year. Motions of fainter stars are also more difficult to accurately determine. But in any event, the greater the angular extent of the ellipse, the closer the star as diagrammed below .
In the above drawing, “A” is an astronomical unit (AU) or the distance from the Earth to the Sun. In modern times this has been measured to great precision and is on average 92,955,807 miles or 149,597,871 km. But since the orbit of the earth is slightly elliptical, the earth-sun distance varies from a farthest point or 1.017 AU at aphelion in July to a nearest point of 0.983 AU at perihelion in January. The difference amounts to about 3.16 million miles. And yes, for those of us in the Northern Hemisphere, we are closer to the sun in winter but then the seasons are caused by the tilt of the earth and not the distance to the sun. And for those emotionally challenged by this absurdity, the Southern Hemisphere has the opposite and expected relationship.
Thus over the course of a year, the target star appears to trace an ellipse in the sky whose diameter spans an angle of 2α as shown below
The equation for the distance can then be simplified to
Where we have made the two assumptions that
A “parsec” is that distance from the earth to star when the angle, θ, is one arc-second and is a distance of 3.26 light years. Note that the parallax angle is inversely proportional to distance.
The actual calculation for one arc-second is
[ (2π) radians / (360) degrees] * (1/60) degree/arc-minutes * (1/60) arc-minutes/arc-seconds = 4.848 * 10-6 radians
so that the distance d2 for that angle is
Switching units, we note one light year is
(365.25) days/year * (24) hours/day * (3600) seconds/hour * (186,280) miles/second = 5.878 * 1012 miles
Despite its straightforward nature, this unit of measure remains confused in the public mind. In the Star Wars movie franchise, the loveable smuggler Hans Solo brags that his ship, the Millennium Falcon, made the infamous “Kessel” run in less than 12 parsecs breaking a long standing record. Attempting to sound both hip and scientific, the writers confused units of distance for those of speed. Bewildered fans, whose lives revolve around movie characters, were only slightly placated when director Stephen Spielberg spun the mistake as referring to the quality of the ship’s navigational computer plotting shorter routes rather than any ability to outrun revenue agents.
We have first made the assumption that because d1 >> d2 then we can set the angle, φ, to zero. We would have preferred to measure the angle the star makes with respect to the sun (90 – α) instead of approximating it with (θ + φ). But we cannot generally observe both during daylight hours. Also the measurement would be difficult since we would have to locate the exact center of the sun to great precision.
We can straightforwardly calculate the error in the estimation of d2 is because of this assumption. To accomplish this we define a few specific lengths as follows
where we have defined the lengths
Where an assumption has been made that φ = 0. We can write
The first step is to use the Pythagorean theorem to calculate several lengths
which can be refined even more using
So that finally
We can apply this to the error estimate giving
And the absolute value of the relative error is
And if we parameterize the distances using
So for reasonable values of the parameters, this assumption is more than reasonable. For instance the nearest star, Proxima Centauri is 4.243 light years or 268,300 AU and the distant background stars thousands of times farther away than that. So for k2 = 268,300 and k1 > 1000, we have
which is well below our best instrumental measurement errors.
We also note that the angle θ is never more than one arc-second which is 4.848 * 10-6 radians. And so a simple expansion of the tangent( θ ) function is as follows
For one arc-second the error associated with this approximation is on the order of one part in 1012 and even less for smaller angles.
From the surface of the earth the blurring of the atmosphere limits the maximum angle we can measure to slightly less than about 0.01 arcseconds for the brightest stars. This corresponds to stellar distances of 100 parsecs or about 320 light years. This proved so difficult that the first measurement was only made in 1838 by Friedrich Bessel of Germany for the star 61 Cygni which has a parallax angle of about 0.287 arc seconds. The closest star Proxima Centauri, has the largest parallax observed of 0.772 arc seconds. Indeed, the total number of stellar parallaxes ever observed from the surface of the Earth is barely one hundred.
This changed in 1989 with the launch of the Hipparcos mission which returned data until 1993 when it consumables were exhausted. From Earth orbit, the distances to about 120,000 of the brighter stars were measured with an accuracy of roughly 1000-2000 micro arc seconds or 1600-3200 light years. An additional one million stars were measured to an accuracy of 10,000 micro arc seconds. There are about 80 million stars within 2000 light years so this is a small fraction of the total.
In July of 2014, the Gaia satellite was launched and expects to measure the parallaxes of one billion stars. The best accuracy will be 6.7 micro arc seconds and a few tens of micro arc seconds for fainter objects. This is roughly one percent of the entire Milky Way Galaxy.
1. The actual motion of the star against the background is a projection of the earth’s orbit onto a plane which is perpendicular to a line drawn from the sun to the star. And the projection of an ellipse is also an ellipse.
2. For simplicity, the target star diagrammed is along a line at right angles to the plane of the earth’s orbit. But without loss of generality, the geometry have the same considerations for any star. Note that if the star were on a line that was entirely along the plane of the earth’s orbit, its apparent path would collapse from an ellipse to a back and forth motion along a line.