Determining the Astronomical Unit

BACKGROUND

The distance from the Earth to the Sun--the Astronomical Unit (AU)--is one of the most important fundamental constants in astronomy and astrophysics, most significantly as the basis of stellar parallax. Parallax is not the only means of measuring distances in space (in fact, since it is dependent on measuring angles it can only measure as far as our best telescopes have angular resolution): out at vast distances we are dependent on objects such as type 1a supernovae and Cepheid variable stars, whose luminosity (and with it, luminosity distance) can be objectively measured due to known properties that they have. However, without the definitive distance measurement offered by parallax, it would be impossible to "calibrate" measurements of such objects--we would be able to tell that there was a correlation between the period and luminosity of Cepheid variables, for instance, but we would only be able to measure their distance relative to one another without the parallax of nearby Cepheids grounding that scale in absolute distance. As such, parallax is the basis of the cosmic distance ladder, and the Astronomical Unit is the basis of parallax.

Parallax uses the relative positions of stars in the sky as the Earth orbits the Sun to determine distances. You can read more about it at the source.

The quest to measure the AU, however, dates back to long before stars were even known to have varying distances. Since nearly all of them relied on parallax of one object or another (the Moon, Sun, or planets) as viewed from different locations on Earth, every estimate until the late 1600s, when the Earth's radius was first accurately measured, was expressed in units of Earth radii. 

Parallax of Venus, measured during a transit of that planet, was used by many astronomers in the 1600s (source).

Our friend Aristarchus, whom we last encountered doing a reasonably accurate job measuring the size of the Moon, attempted to do so using the angle between a quarter moon and the Sun. Eratosthenes was the next to try, and--once again, depending on which type of stadion he was using--he either dramatically underestimated the distance or came within a few percent. In any case, the definitive value for the AU was, for many centuries, the one measured by Ptolemy using lunar parallax, which was only about one twentieth of the reality. Despite being so far off the mark, his value was more or less universally accepted until Kepler called it into question in the 1600s. Not long after, astronomers observing a transit of Venus from different locations on Earth attempted to measure the AU using solar parallax, pushing the generally accepted value of the AU to about half of what we now know it to be. The first relatively accurate measure of the AU finally came in 1672 at the hands of the astronomers Cassini and Richer (Christiaan Huygens produced a good estimate of it in 1659, but it seems he just got lucky--his methods were very flawed). Finally, more advanced methods using the speed of light, radar-gauged distances to Venus and Mars, and telemetry from space probes have continued to refine our knowledge of the AU--three additional decimal places of precision were added in 2009, giving us the AU to within $3 \times 10^{-7}$% uncertainty.

CALCULATIONS

Lacking a transit, interplanetary radar, or a space probe, we departed from historically treaded ground in order to measure the AU by new means. All of the equipment and direct observations we performed can be read about in my discussion of how we measured the rotational period, angular size, and rotational velocity of the Sun. With these constants in hand, it is simply a matter of combining them in the right way to reach a measurement of the AU.

Our first step was to determine the physical size of the Sun. Knowing the rotational period and velocity of the Sun at the equator, this became a simple problem of $D = vt$. Our distance $D$ was the circumference of the Sun, while the rate $v$ was the rotational velocity and time $t$ the rotational period, $p_\odot$. Since what we're really after is the radius of the Sun, we can plug $2 \pi r_\odot$ in for $D$ and solve for $r_\odot$, giving us $r_\odot = \frac{vp_\odot}{2 \pi}$. We'll plug in the actual values later--for now, it's more important to have everything in terms of known constants.

Next it was time to combine the radius and angular size in a relation involving distance. This was a simple matter of drawing a right triangle, with legs going from the center of the Earth to the center of the Sun and from the center of the Sun to the edge of the Sun, and a hypotenuse going from the edge of the Sun back go the Earth. The angle closest to Earth is the angular size of the Sun (which we'll call $\alpha$), the leg going across the Sun's face is its $r_\odot$, and the leg going from the Earth to the Sun is the AU.

If the circle on the right is the face of the Sun, $d$ is its diameter, $\delta$ is its angular size, and $D$ is the AU (source). Our triangle has legs $D$ and $d/2$, and an angle of $\delta / 2$ at the leftmost vertex (source).

These three quantities--two of which we know--can be related using trigonometry. Thanks to SOHCAHTOA, we know that the tangent of an angle is opposite/adjacent--in the case of $\alpha$, $r_\odot / 1$ AU. However, to get the exact proportions of our triangle right we need to divide $\alpha$ by two. Thus, 1 AU = $r_\odot / tan(\alpha)$. Plugging in for $r_\odot$ gives us 1 AU = $\frac{\frac{vp_\odot}{2 \pi}}{tan(\alpha / 2)}$.

Now, finally, let's plug in the constants we measured in the last three posts. $p_\odot$ we have as 26.3 days, $\alpha$ as 0.56 degrees, and $v$ as 0.9597 km/s. Thus,\[1 \: \: \textrm{AU} = \frac{\frac{(0.9597 \: \: \textrm{km/s})(26.3 \: \: \textrm{days})}{2 \pi}}{tan(0.28 \: \: \textrm{degrees})} = 7.1 \times 10^7 \: \: \textrm{km}\]

CONCLUSIONS

We entered into these calculations aware that we were carrying in a significant amount of error from our past measurements, particularly from our measurement of $v$, the rotational velocity of the Sun. However, given that we knew that value to be just under half of the literature value, it then meets our expectations to see that our result of $7.1 \times 10^7$ km is about 47% of the current literature value for the AU. While the error affecting our earlier observations and calculations of $p_\odot$, $v$, and $\alpha$ was significant, particularly in the case of $v$, it seems that our mathematical methodology here has introduced no further error. Thus, if the component observations were to be repeated in such a way as to minimize the error we encountered, it is likely that repeating the calculations above would provide an accurate value of the AU. Furthermore, while a factor of two is quite large for measuring distances within the solar system (if NASA were to send a probe to the Sun assuming it was half as far away as it really is, they'd be in trouble), a factor of two would land more significant measures of distance using parallax and other cosmic distance rulers comfortably within an order of magnitude of the reality. While our experiment here was not wholly successful in producing an accurate measure of the AU, future iterations of it, adjusted to account for the error we encountered, should be able to accurately gauge the AU using a novel methodology.