Equipped with a home-made astrolabe and Kepler's laws, one can arrive at accurate estimates of the orbits of planets. Over the past three years, I have used an astrolabe to collect sightings of celestial bodies visible from my backyard. This paper shows that it is possible to measure one's latitude and longitude, the current time, locations of celestial bodies, and the sizes of the orbits of nearby planets. When the astrolabe is used to collect both daytime and nighttime observations of Venus, one can also demonstrate that Venus orbits the Sun.

Even living in the midst of city lights, you can still see some of the brightest stars and planets with the naked eye. Over a span of two or three years, you can watch the planets move across the sky against the background stars. Surprisingly, with basic observations of these objects and a simple tool called an astrolabe you can tell the time, find compass directions, determine the season, and determine the diameter of planetary orbits.1,2 The astrolabe is based on a stereographic projection of the stars, in which they are projected onto a plane that rotates around the celestial poles. Astrolabes have been made since antiquity, and you can make your own.3–6 

In the hands of a skilled navigator, an astrolabe was often the tool of choice, yielding elevation sightings accurate to better than a degree.7–10 Additionally, the best source of time until the 20th century had been obtained from the rotation of the Earth, which can be measured by tracking the movements of stars. There is also ample evidence that astrolabes were used for teaching astronomy.11 It is, therefore, natural to ask if one can use an astrolabe to measure the solar system.

Over the past three years, I have used an astrolabe to collect sightings of celestial bodies visible from my backyard, hoping to learn what could be measured. My astrolabe follows a design popular in medieval England and which is described by the famous poet Geoffrey Chaucer in his Treatise on the Astrolabe12 in a letter to his son.13 As shown in this article, the level of accuracy obtained by an astrolabe is sufficient to measure planetary orbits if one uses the modern expedient of statistical analysis. Moreover, one can supplement nighttime observations of Venus with daytime observations aided by binoculars. Astrolabe measurements of these combined sightings suggest that Venus orbits the Sun.

The design of my astrolabe is explored in Sec. II, and its use is outlined in Sec. III. Further usage instructions are in the supplementary material associated with this article. With these basics in hand, Sec. IV provides several possible classroom lessons or projects that can be undertaken using the astrolabe. Section V describes a dataset collected with my astrolabe. The remaining sections explore possible student projects that can be done with the dataset: Sec. VI uses the dataset to measure the accuracy of sightings collected with the astrolabe, Sec. VII demonstrates how the sightings can be used to measure planetary orbits, and Sec. VIII suggests some avenues for additional projects.

The main use of an astrolabe is to measure the elevation angle above the horizon of celestial bodies, and from these sightings, derive other quantities of interest. To determine planetary orbits, one needs only a set of observations containing the local time, date, and the right ascension of each planet. These quantities can be derived from elevation sightings using the astrolabe as a mechanical calculator.

The front of the astrolabe consists of two pieces pinned together with a screw as shown in Fig. 1. The main body of the instrument is usually held stationary and contains etchings of elevation contours, which are drawn using stereographic projection. Since the north celestial pole's elevation depends on one's latitude, the elevation contours on the astrolabe also depend on one's latitude; compare the instruments shown in Fig. 2.

The back of the main body contains a scale for taking elevation sightings. My astrolabe also contains a plot of the equation of time for translating between local apparent solar time and local mean solar time. These two scales are shown in Fig. 3.

The stars are etched on a clear sheet of plastic known as the star chart (Fig. 4); this is free to rotate over the main body and shows the locations of several bright stars.

On the star chart, the right ascension and declination coordinates of each star are represented using polar coordinates via stereographic projection. The right ascension is indicated around the circumference of the star chart in hours with each hour being equivalent to 15°.

Circles of constant declination are concentric with the center screw. In an astrolabe for the northern hemisphere, the north celestial pole at declination 90° is placed at the center screw. Stars with positive declinations (for example, Capella) appear inside the celestial equator on the chart, while those with negative declinations (for example, Sirius) appear outside the celestial equator. For a southern hemisphere astrolabe, the south celestial pole (declination 90°) is at the screw, and the stars would appear in different locations on the chart as a result.

Because the star chart is drawn with stereographic projection, angles (but not areas) are preserved. This makes it possible to identify certain constellations on the astrolabe, because their stars are in their correct positions relative to each other. Circles in the sky not passing through the south celestial pole appear as circles on the star chart. Because of this, the ecliptic plane defined by the Earth and the Sun appears as a circle drawn on the chart whose center is offset by roughly 23.4° from the celestial pole.14 

The Sun's right ascension increases at a mean rate of approximately (1/365.25) degrees per day, so it completes a counterclockwise circuit on the star chart once per year. The chart shown in Fig. 4 has a calendar scale plotted around its circumference that allows the user to place the Sun at its proper location along the ecliptic for any date. This is explained in further detail in the supplementary material.

I live in a light-polluted suburb of Washington, DC, from which only a few bright stars are visible.15 After a few sessions of observing, I determined that 12 frequently visible stars could serve as “guide stars”; these are the ones plotted on the star chart of my astrolabe (Fig. 4).

Determining star positions in right ascension and declination has been an important task for astronomers since antiquity. All 12 of my guide stars have published right ascension and declination coordinates from the European Space Agency's Hipparcos mission. Although these data are substantially more accurate than necessary, they are formatted in a way that makes drawing the star chart with a computer very easy.16 If these high-quality data were not available, one could resort to using the astrolabe itself to make these measurements; Chaucer gave detailed instructions for this task (Ref. 13, Sec. II and Refs. 17 and 18).

The astrolabe is a capable instrument, but it is not completely straightforward to use. While the reader is encouraged to read Chaucer's Treatise in its entirety, this article outlines the basic tasks necessary for measuring the orbits of planets. Some of the tasks that one needs to accomplish depend on others, as shown in Fig. 5. These are discussed in more detail in the supplementary material.

Here are a few hints that the aspiring astrolabe user should consider:

  • Practice sighting objects of known dimensions during the daytime before attempting nighttime sightings. The author's video demonstrates the basic procedure.17 

  • Make sure that the astrolabe hangs freely from its top loop. Rather than hanging the astrolabe's ring from one's finger, hang the astrolabe from a loop of string passed through the ring.

  • Adjust the tension on the center screw so that the sighting pointer does not drift after being set. Once you have the pointer aligned with an object, make sure to avoid bumping the pointer until you have read it.

  • Avoid sighting during windy conditions.

  • Wait for the instrument to stop moving between each adjustment of the sighting pointer, and make only small corrections to the pointer as you sight an object.

  • Do not look directly at the Sun. Instead, cast a shadow with the astrolabe and its pointer.

  • Take a few sightings of an object and average them to reduce the elevation error.

  • Take sightings of as many guide stars as possible, then attempt to “split the difference” across their locations on the astrolabe (Ref. 13, Sec. II.5).

  • Sighting Venus during the daytime is a challenge. Consult the author's video for detailed instructions.18 

The astrolabe incorporates an elevation sighting scale on its back (see Fig. 3). Taking a sighting is basically the same as doing so with a theodolite, sextant, or similar instrument. First, locate the object you wish to sight. Use your right hand to hold the astrolabe from a loop of string passed through the thumb ring at the top, so that the back face of the astrolabe is on your left. Plant your feet firmly on the ground facing the object, holding the astrolabe at eye level. When the astrolabe stops swinging, carefully align the main body of the instrument and back pointer so that the object “just skims” the top edge of the pointer. Then you can read the elevation of the object from the outermost scale on the back of the astrolabe.17 

Amusingly (to me, anyway), Chaucer concluded his brief instructions on how to take a sighting by saying “This chapitre is so general ever in oon, that ther nedith no more declaracioun; but forget it nat” (Ref. 13, Sec. II.2). Loosely translated, this can be expressed as “We'll be using this so frequently that I won't bother to mention taking elevation sightings again, but don't forget how to do it.” This may be true, but taking accurate elevation sightings required more practice than I expected!

Even if one does not learn anything else about the astrolabe, taking elevation sightings is a useful navigational skill since it can be used to determine one's latitude. To the accuracy of my astrolabe (which is about 2°3°, according to Table II in Sec. VI A), one merely needs to sight the elevation of Polaris.

Interestingly, Polaris was about 4° away from the north celestial pole during Chaucer's time, yielding an angular difference that is detectable with my astrolabe. Chaucer describes three options to deal with this [Ref. 13, Sec. II and Refs. 23–25], each of which is a rather more elaborate process than sighting Polaris. These options are still useful when locating the north celestial pole using a telescope.

Astrolabes have been used for teaching basic astronomical concepts since antiquity, and using astrolabe-based data in the classroom opens many pedagogical opportunities. A few are described in the following paragraphs.11,19

There are plans for complete, ready-to-use astrolabes from a variety of sources.3,4,20,21 For timekeeping activities, laminated paper works well as a material for the main body, and a printable overhead transparency can be used for the star chart. A folded laminated strip of cardboard attached with a brass fastener can serve as the pointer. An instructor can print a set of plans before a class, during which students can construct their astrolabes. After students have completed their astrolabes, it is a good exercise to “try them out” using the instructions in Sec. III. If Polaris is visible, sightings of its elevation should closely match one's latitude.

For projects requiring more extensive data collection, sturdier materials, such as wood or laser cut plastic,22 are preferable. My first astrolabe consisted of printed sheets of paper glued to thin plywood; these can be made without access to a sophisticated workshop.23 The dataset described in this article uses two sizes of 1/4 in. thick plastic astrolabes roughly 4 in. and 8 in. in diameter. Students at my university have successfully constructed their own laser cut astrolabes from my plans with limited instructor supervision.

The center pin of a wooden or plastic astrolabe should be what engineers call a “location fit.” There is no side-play in the pin, but it stills allows rotational movement of the star chart and pointers. My plans are configured so that a US standard 6–32 machine screw gives location fit with no further intervention. After three years of near-daily use, there is no visible wear on any part of the instrument except for some scratches on the star chart. If a different size screw is desired, a good fit can be obtained by making the center holes initially a little too small. After they are cut, the holes can be carefully widened until the screw just passes. The classic clock maker's tool for this job is called a “broach,” though the reader can certainly improvise an alternative tool.

For instructors not wishing to manage the construction of astrolabes, the dataset presented in this article provides numerous opportunities to teach students about statistical properties of measurement errors. The entire dataset and all source code for running the analyses described in Secs. VI and VII are available from the author's public GitHub repository.24 The author has used these data to illustrate important statistical distributions (particularly the normal distribution) in elementary classes. Students can also analyze the statistics of timekeeping errors, as described in Secs. VI A and VI B. Several kinds of errors are present in the data. By considering the steps shown in Fig. 5, students can devise analyses to isolate these errors from each other.

The astrolabe is an ideal way to teach timekeeping. Students would need to master the skills leading up to “Reading the local apparent solar time” in Fig. 5 with the skills downstream of that point being optional. These skills are explained in the supplementary material. Laminated paper astrolabes are sufficient for timekeeping. Since taking sightings requires some practice (and clear weather), a minimum of two class meetings should be devoted to developing this skill. Once these skills are mastered, determining the time can be completed in a few minutes.

Historically, printing errors were substantial,10 but they should be minimal with a modern printer or laser cutter. Students can estimate the other kinds of errors in their sightings using the techniques described in Sec. VI.

Measuring the orbit of Venus is a feasible activity given about a month of class meetings. It does require preplanning to ensure that students are able to collect sightings of Venus near its furthest apparent distance in the sky from the Sun, that is, when it is at maximum solar elongation. Not every semester would present such an opportunity. In order to collect the data (timeseries of the right ascensions of Venus and the Sun), students must master the steps leading up to “Computing right ascension” in Fig. 5. Although laminated paper astrolabes might work for this task, those made with more durable materials are more accurate and easier to use. Once the data are collected, the analysis described in Secs. VII A or VII B will allow the class to derive estimates of Venus's orbital diameter.

Once a student has mastered the astrolabe, the only thing preventing them from measuring orbits of the other planets is time. My dataset contains just over three years of sightings. As Sec. VII shows, this is certainly sufficient and provides some margin for error and missing data.

Using astrolabes I fabricated, I carried out 773 observing sessions over the course of three years. Most of the observations were collected with two laser-cut plastic astrolabes, a smaller one with a 4 in. diameter and a larger one with an 8 in. diameter. I made 2093 distinct elevation sightings as summarized in Table I. In each session, I recorded the current time from my computer's clock, the elevations of all visible planets, and the elevations of all visible guide stars. I avoided taking sightings during windy conditions to minimize errors. Although it would have been preferable to take sightings from the same location, my view of the sky is obstructed by several large trees. I, therefore, had to move to various locations in my yard (roughly a square 50 m on a side) in order to collect all sightings for a given observing session. The reader is encouraged to examine the data and the associated analysis script, which was used to draw many of the figures in this article.24 

With the exception of the daytime sightings of Venus, I used no optical aid other than the astrolabe. Venus can be seen during the daytime without optical aid under ideal conditions. This requires knowing where to look first, for which binoculars are helpful. Once acquired, the elevation angle to Venus can then be sighted with the astrolabe. Since this is a somewhat challenging procedure, I have documented it in a video.18 

Before embarking on computing the orbits of planets, it is useful to determine the accuracy of the measurements one can collect. Since the process for deriving time and right ascension measurements involves several steps, it is important to check the accuracy at several points along the process shown in Fig. 5.

Using known positions of the Sun and stars, it is an exercise in spherical trigonometry to translate these positions into elevation angles at any given location and time on the Earth. These values can be compared with the elevation sightings collected with the astrolabe. The elevation errors appear to follow a normal distribution (see the supplementary material), which indicates that the errors are not skewed. The overall statistics are summarized in Table II.

A two-sample t-test from Table II yields p =0.005, so there is a significant difference in mean elevation estimation error for the Sun versus the stars. In short, the Sun estimates are a little more centered than those of the stars. I suspect that this difference has to do with the design of my particular instrument and the techniques I used. To sight a star, the user must “skim” the star along the top edge of the pointer. This tends to bias the estimates lower in elevation, because an overestimate means that the pointer obstructs view of the star. To measure the Sun, one casts a shadow with the pointer and tries to minimize its apparent size. The procedure for measuring the Sun appears to have less elevation bias than directly sighting a star.

A related instrument, the mariner's astrolabe, is optimized for sighting the Sun. The pointer on the mariner's astrolabe has two vanes, each perforated by a tiny hole through which the sunlight may pass only when the correct elevation is displayed. The mariner's astrolabe can be used to obtain substantially better accuracy than what I was able to collect, typically 0.1° for skilled users on land.7–9 

In what follows, sightings obtained with both my four and eight-inch astrolabes are pooled. One would generally expect that the larger instrument would give noticeably better elevation accuracy, since the elevation scale is larger. However, I found no statistically significant difference in accuracy between them. I suspect that this effect is caused by the straightness of the sighting pointers. Over the course of three years, the larger astrolabe's sighting pointer has become a bit more warped than that of the smaller one.

The local mean solar time can be compared with my zone time (Eastern Standard Time). Using either the Sun or stars results in n =765 distinct time estimates; the remaining 773–765 = 7 observing sessions are instances where not enough data were collected to uniquely determine the time. For these estimates, the distribution is close to normal; the mean difference was—7.4 min and the standard deviation was 57 min.

The length of a daytime observing session is quite short (typically, one or two elevation sightings), while nighttime sessions are typically longer. It would not be surprising if there were a difference in the time error for day vs night observing sessions. Moreover, there is a significant difference between elevation errors for the Sun vs the stars (see Sec. VI A). However, in the face of these potential problems, an analysis of variance (ANOVA) test shows no significant difference between time estimates derived from the Sun in the daytime vs the stars at night.

By international agreement, UTC is the local mean solar time at 0° longitude.25 Longitude is, therefore, obtained by subtracting the local mean time determined from the UTC time and converting to degrees of longitude by multiplying by 15° per hour. My computer's clock provides an accurate common time reference, which is aligned to the local mean solar time at 75° West longitude. The difference of –7.4 min noted above translates to a longitude of 76.85° West, which is, therefore, an estimate of my home longitude. This is about 13 km off from the true value, which is close to 77° West.

In Sec. VII, we will use the time series of right ascension measurements of a planet to estimate its orbit. Therefore, it is important to measure the accuracy of the right ascension measurements obtained with the astrolabe.

In addition to the right ascension measurements of each planet, I also predicted the expected right ascensions with a computer algorithm based on Kepler's laws in combination with modern values of the orbital elements.26 Each right ascension obtained using the astrolabe can be compared with these expected values; the discrepancies are summarized in Table III.

Chaucer claimed (Ref. 13, Sec. II.3) that right ascension errors can be noticeably worse when an object is nearly due south; this is because the elevation contours drawn on the astrolabe become tangent to the object's path through the sky. For instance, in Fig. 6, the impact of a ±2° elevation error appears to depend significantly on whether the star is near due south (the vertical line of symmetry of the astrolabe) or not. Chaucer cautioned readers that

But natheles, in general, wolde I warne thee for evere, ne mak thee nevere bold to have take a iust ascendent by thyn Astrolabie, or elles to have set iustly a clokke, whan any celestial body by which that thou wenest governe thilke thinges ben ney the south lyne.

In general, I warn you not to measure a right ascension of a celestial body using your astrolabe—or use it to set your clock—if it is near due south.

This claim can be tested using my data. Figure 7 displays right ascension error as a function of azimuth for the planets. While there is a noticeable discontinuity at due south, the error does not change much as due south is approached from either side. While the largest errors—which may be considered outliers—do seem more pronounced near due south, I think that Chaucer was being overly cautious.

The design of Chaucer's instrument was quite similar to mine, though he was based in Oxford, England at latitude 51°50. Oxford is substantially further north than my home latitude of 39°, which makes the tangency between elevation and declination contours more severe; notice the difference in contours shown in Fig. 2. It may be that Chaucer's advice is sound in his location, especially when the instrument is in the hands of a beginner.

I believe that an explanation for the discontinuity in Fig. 7 at azimuth 180° could be due to buildings that blocked my view of the sky. The walls of my house are closely aligned with north/south and east/west, so it is very easy to tell if an object has passed due south by simply sighting along one of the outer walls of the house. If an object appears east of due south (so its azimuth is less than 180°), I am apparently less likely to check if this is truly the case. Such an error (mistakenly choosing a right ascension that is currently east as opposed to west) would cause its right ascension error to be positive.

From my location I was able to observe Venus, Mars, Jupiter, and Saturn (see Fig. 9 and Table III). Although Mercury is certainly visible without optical aid, its maximum elevation is too low to be consistently observed from my location. I made no observations of Uranus or Neptune; both are difficult targets without a telescope and were unknown to Chaucer.

Knowing the right ascension of a planet is not sufficient to place it in its true location in space, nor can it be used to determine an orbit without further information. The key insight that allows one to make the leap from right ascensions to orbital dimensions is provided by Kepler's laws, which connect orbital dimensions to orbital periods. The latter can be derived from a time series of right ascensions, as explained below.

Kepler's third law27 states that if r is the semi-major axis of the orbit and T is its orbital period

r3=kT2,
(1)

where k >0 is a constant of proportionality, which depends on the mass of the Sun and the Newtonian gravitational constant G.

Directly measuring the orbital period of a planet is complicated by the fact that both the planet and the Earth are in motion. Fortunately, the planet's orbital period can be estimated using a geometric trick that works because the planets lie close to the ecliptic. There are distinct times when the Sun, the Earth, and the planet are (nearly) colinear in space. These times are called solar oppositions or solar conjunctions, depending on whether the planet and the Sun are on opposite or the same side of the Earth, respectively. The path of a planet and the Earth between two consecutive solar oppositions or conjunctions is shown in Fig. 8. The time interval between two consecutive conjunctions or oppositions is called the synodic period of the planet.

Kepler's second law states that if a planet's orbit is circular, then it will move at a constant angular speed over its entire orbit. All four planets I observed have orbits that are close enough to circular to assume that this is so.

Over one synodic period T (days), the angular distance traveled by the Earth is approximately

2πT365rad.

Over the same period of time, a planet outside Earth's orbit will traverse only

2πT365365rad.

Let us use the Earth's orbit as a yardstick for both radius and period. That is, for the Earth, r =1 astronomical unit (AU) and T =365 days, so that

k=(1/365)(AU3/day2).

Therefore, Kepler's third law asserts that for an outer planet (one whose orbit lies outside Earth's orbit), the semi-major axis r in astronomical units is given by

r=(TT365)2/3AU.
(2)

The situation is reversed for an inner planet. In this case, the planet never exhibits solar oppositions since the planet cannot be on the opposite side from the Earth as the Sun. As a result, the planet oscillates between solar conjunctions on opposite sides of the Sun. The synodic period T is the time in Earth days between two of the same kind of conjunction. This leads to

r=(TT+365)2/3AU.
(3)

The synodic period of a planet is the key to unlocking its orbital dimensions and can be derived from time series of right ascensions. Because the synodic period relates the Earth's and the planet's orbits to each other using conjunctions or oppositions, it helps one to subtract the planet's right ascensions from the Sun's right ascensions. The resulting differences, called solar elongations, have the convenient property that 0 h correspond to conjunctions and ±12 h to oppositions.

Figure 9 shows the solar elongations of each of my planetary sightings over my entire dataset. One thing that is immediately apparent is that Venus never appears in solar opposition in my data. Especially taking into account the daytime sightings, it appears on both sides of the Sun. Therefore, one would have expected that if they were possible, solar oppositions of Venus would be present. We know that Venus is an inner planet, because it never appears “outside” Earth's orbit. Moreover, a hint of sinusoidal variation is visible for the Venus sightings in Fig. 9. Thus, it seems reasonable to claim that I have directly observed Venus orbiting the Sun instead of the Earth. Unassailable proof of this is credited to Galileo's telescopic observation of the phases of Venus in time with this sinusoidal variation. This does not establish that the Sun is at the center of the solar system, only that Venus orbits the Sun. Because the other three planets exhibit solar oppositions, we must conclude that they are outer planets.

The synodic period is measured between consecutive oppositions or conjunctions, though it is clear from Fig. 9 that I did not observe many of these events directly. Fortunately, the synodic period can also be estimated by considering the time interval between recurrences of a solar elongation. That is, the synodic period is the time between consecutive crossings of an arbitrary horizontal line in Fig. 9.

As an example, I saw Saturn on 6 October 2020 at solar elongation 6.1 h, and again at that same solar elongation on 23 October 2021. Both of these observations correspond to points in Fig. 9. Together, these two observations suggest a synodic period estimate of 382 days. Using this as the synodic period results in an estimate of 7.96 AU for Saturn's semi-major axis. Over the entirety of my data, there are many such possible recurrences.

Although a given solar elongation may not recur exactly, there are many cases where a similar solar elongation occurs later in the data. To capture these “approximate recurrences,” solar elongations were grouped into 2-h bins. Each pair of observations in each bin corresponds to a possible approximate recurrence of that solar elongation, and the time interval between them is then an estimate of the planet's synodic period. Figure 10(a) shows the distribution of these estimates for the synodic periods.

Because binning also collects observations of the same opposition or conjunction rather than a recurrence, short intervals must be removed. Intervals greater than 200 days between observations in these bins correspond to instances where the planet had re-appeared in the same relative location and were used. All of these time intervals between pairs of observations in these bins were aggregated into a distribution, shown in Fig. 10(a). Each of these time intervals was used to estimate the orbital semi-major axis using Eq. (2) (for the outer planets) or Eq. (3) (for Venus). The resulting estimates are shown in Fig. 10(b) and Table IV. Note that the counts in the n column of Table IV correspond to pairs of sightings, and so differ from the number of sightings of each planet in Table III.

Because I collected numerous observations of Venus on both sides of the Sun, we can cross-check estimates of its orbit by another method that does not rely on Kepler's laws. This is to use geometry to directly estimate the orbital radius from Venus's largest solar elongation, if we assume that its orbit is circular. The geometry of this situation is shown in Fig. 11(a), where θ is the maximum elongation. We can read off the maximum values of solar elongation directly from Fig. 9.

The geometry in Fig. 11(a) indicates that we only need to estimate the maximum solar elongation. When this maximum occurs, the angle between the Sun-Venus and Earth-Venus segments is a right angle. (If a telescope is available, Venus will appear as a half-disk at this time.) If there were no errors, the distribution of these observations would have a single sharp peak at the maximum solar elongation with no observations larger than this value. To estimate this sharp peak, we can simply use every observation of Venus to derive an estimate of Venus's orbital radius by taking the mode of the resulting distribution. (Taking the maximum value of solar elongation has the unfortunate consequence of also selecting the largest solar elongation error. The mode tends to be less impacted by the errors.)

Figure 11(b) shows the distribution of estimates obtained by computing the sine of the solar elongation of Venus, in which a total of n =47 observations were available. The distribution of these radius estimates has a fairly heavy left tail, corresponding to the points in time where Venus was not at its maximum apparent distance from the Sun. This tail is not very relevant, since we are most interested in the maximum value represented. To estimate maximum, we can use the mode, which is 0.650 AU. The standard deviation for the observations is 0.131 AU. The true value of 0.723 AU is within one standard deviation of 0.650 AU, so we conclude that using geometry alone yields a good estimate of Venus's orbital radius.

The data I collected24 can be used to answer many other questions beyond those described here. For instance, I also collected sightings of the Moon, including its phase. Can these determine the Moon's orbit closely enough to be able to predict its location in the future? How does the error depend on the forecast length?

There is also the possibility of using a telescope in conjunction with the astrolabe to collect measurements of fainter objects. Using a telescope, I have already collected a few observations of the phases of Venus. Even though they cannot be seen by the unaided eye, even a small telescope allows one to track the planets Uranus and Neptune. I have seen Uranus through a telescope a handful of times and perhaps even Neptune, but not enough to perform any systematic analysis.

If one is especially ambitious, the techniques explained in this article could be made to work for minor planets such as Vesta or Ceres. Some of the minor planets are sufficiently close to the ecliptic that the techniques here can work without modification. For minor planets whose orbital planes are not as closely aligned with the ecliptic as the major planets, one could use Chaucer's instructions to determine their declinations (Ref. 13, Sec. II.30). With these in hand, Kepler's laws could help predict the orbits of minor planets. This might be a challenging and rewarding exercise to attempt.

Finally, one might consider several other studies in how the astrolabe's performance varies with changes to either its construction or its usage. For instance, it stands to reason that larger astrolabes might yield more accurate sightings. One could fabricate several instruments from a few inches in diameter to nearly a meter and collect a few months' worth of sightings to compare them.

Table IV affirms that given enough data (a few years' worth), an astrolabe can help you measure the solar system, even if you do not have dark skies. I have demonstrated that it is possible to measure:

  1. one's latitude and longitude to around 1°,

  2. the time to within 30 min consistently, often much better,

  3. the current locations of objects in the sky, and

  4. the size of the orbits of the nearby planets.

Based on the observations of Venus, you can even suggest that some planets orbit the Sun, and not the Earth. When supplemented with daytime observations, the solar elongations of Venus indicate that no oppositions ever occur, a clear signal that it orbits the Sun. Admittedly, to make the daytime observations of Venus I had to use binoculars to “acquire” the sighting, even though the astrolabe was used to measure the elevation. This is perhaps an overly modern advantage.

Even without an astrolabe, the reader should go outside at night and look at the night sky. It is beautiful and there are many things to be learned.

The author would like to thank the directors of American University's Design and Build Lab, Kristof Aldenderfer and Gustavo Abbott for their assistance in developing the laser cut astrolabes used to collect the data described in this paper. The author would also like to thank the anonymous reviewers for their helpful feedback on this article.

The author has no conflicts to disclose.

All of the data and analyses described in this article are freely available.24,31 The reader is also encouraged to make their own astrolabe. Plans for the astrolabes used to collect the data in this article are freely available.3,4

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N.
de Hilster
, “
Observational methods and procedures for the Mariner's astrolabe
,”
Mariner's Mirror
100
,
261–281
(
2014
).
8.
W.
Köberer
and
F. A.
Main
, “
On the attempts to assess the accuracy of the astrolabe
,”
Mariner's Mirror
100
,
198–203
(
2014
).
9.
R.
Knox-Johnston
, “
Practical assessment of the accuracy of the astrolabe
,”
Mariner's Mirror
99
,
67–71
(
2013
).
10.
A.
Chapman
, “
A study of the accuracy of scale graduations on a group of European astrolabes
,”
Ann. Sci.
40
,
473–488
(
1983
).
11.
C.
Eagleton
, in
The Whipple Museum of the History of Science: Objects and Investigations
, edited by
J.
Nall
,
L.
Taub
, and
F.
Willmoth
(
Cambridge U.P
.,
Cambridge
,
2019
), pp.
33
54
.
12.
Although Chaucer is primarily known now as a master of poetry and fiction, his Treatise is apparently the earliest known technical manual written in English! (Ref. 28). It is well organized and is written in clear, technical prose. With a little effort, it can be understood by a modern reader in its original form, especially if one refers to an actual astrolabe.
13.
G.
Chaucer
,
A Treatise on the Astrolabe
(ca. 1400) (
Wentworth Press
,
Sydney
,
2016
).
14.
15.
My observing location tends to be either 8 or 9 on the Bortle scale (Ref. 29), where 1 is the darkest sky and 9 is the most light-polluted sky.
16.
D.
Nash
, “
HYG-database
,” <https://github.com/astronexus/hyg-database> (
2019
).
17.
M.
Robinson
, “
How to use an astrolabe in the daytime
,” <https://www.youtube.com/watch?v=tt-_gkmx8dk> (
2020
).
18.
M.
Robinson
, “
Finding Venus during the daytime
,” <https://www.youtube.com/watch?v=n1vhfexy0ys> (
2021
).
19.
D.
de Solla Price
, “Philosophical mechanism and mechanical philosophy,”
Annali dell'Istituto e Museo di storia della scienza di Firenze
5
,
75
85
(
1980
).
20.
R.
Wymarc
, “
The astrolabe project
,” <https://www.astrolabeproject.com/> (
2019
).
21.
G.
Zotti
, “
Tangible heritage: Production of astrolabes on a laser engraver
,”
Comput. Graph. Forum
27
,
2169
2177
(
2008
).
24.
M.
Robinson
, “
Astrolabe data and analysis
,” <https://github.com/kb1dds/astrolabe_analysis> (
2022
).
25.
To within the accuracy of my astrolabe, anyway.
26.
J. S.
Trefil
,
Space Atlas: Mapping the Universe and Beyond
(
National Geographic Books
,
Washington, DC
,
2012
).
27.
The derivation of Eq. (1) is quite well-known in the literature (for instance, see Ref. 30 for a very elementary derivation).
28.
C. S.
Lipson
, “
Descriptions and Instructions in Medieval Times: Lessons to be Learnt from Geoffrey Chaucer's Scientific Instruction Manual
,”
J. Tech. Writing Commun.
12
,
243–256
(
1982
).
30.
F.
Mošna
, “From Newton to Keppler: One simple derivation of Kepler's laws,”
Italian J. Pure Appl. Math.
32
,
393
–400 (
2014
), https://ijpam.uniud.it/online_issue/201432/35-mosna3.pdf.
31.
See the supplementary material at https://www.scitation.org/doi/suppl/10.1119/5.0097302 for detailed instructions for using the astrolabe, further statistical analyses, and astrolabe plans.

Supplementary Material