Estimating the Scale of the Solar System with Seventeenth-Century Astronomy
By the late seventeenth century, astronomers possessed a remarkably accurate geometric description of the Solar System. The heliocentric model had become widely accepted, Kepler's laws described planetary motion, and telescopic observations had revealed the phases of Venus and the moons of Jupiter. Yet one fundamental quantity remained uncertain: the actual size of the Solar System.
Astronomers knew the relative proportions of planetary orbits, but they did not know the physical length corresponding to those proportions. The Earth–Sun distance, now called the astronomical unit (AU), had not yet been measured accurately.
This essay presents a hypothetical method by which a Galileo–Newton era astronomer could estimate the AU using only knowledge available at the time, together with a single physical assumption: that Venus is approximately the same size as Earth.
The purpose of the exercise is not to reproduce the historical determination of the AU, but to explore how much could plausibly have been inferred using seventeenth-century observations, geometry, and physical reasoning.
Step 1: Choosing a Physical Length Scale
To convert angular measurements into distances, some physical length must be known or assumed.
Suppose an astronomer assumes that Venus has approximately the same diameter as Earth:
The Earth's diameter was already known to reasonable accuracy through geodetic measurements ultimately derived from methods dating back to Eratosthenes. A value close to
would have been entirely plausible.
We therefore adopt
as our assumed diameter for Venus.
Step 2: Observing Venus Throughout Its Cycle
A careful astronomer would observe Venus repeatedly over many months rather than relying on a single measurement.
Near inferior conjunction, Venus appears largest. Near superior conjunction, it appears smallest. At intermediate positions it exhibits intermediate apparent diameters.
Suppose a series of observations yields representative apparent diameters such as:
These values should not be regarded as precise measurements. Seventeenth-century telescopes suffered from chromatic aberration, diffraction, imperfect optics, and atmospheric distortion. Uncertainties of several arcseconds would be expected.
Nevertheless, the observations reveal a consistent pattern: Venus appears roughly six times larger at its nearest than at its most distant.
Since apparent diameter is inversely proportional to distance,
the ratio of distances is approximately
A practical complication is that Venus does not appear fully illuminated throughout its cycle. Near inferior conjunction it appears as a thin crescent, making accurate measurement of its diameter considerably more difficult than the simple geometric model suggests. This would likely be one of the dominant sources of observational uncertainty.
Step 3: Using Kepler's Laws to Determine the Relative Orbit Sizes
Long before the absolute size of the Solar System was known, Kepler's Third Law allowed astronomers to determine the relative sizes of planetary orbits.
The orbital periods of Venus and Earth were known to be approximately
Kepler's Third Law states that
Substituting the observed periods gives
Thus Venus's orbit is known to be approximately 72% the size of Earth's orbit.
The observed variation in Venus's apparent diameter is broadly consistent with this geometry, providing a useful check on the assumptions of the model.
Step 4: Estimating the Astronomical Unit
The assumption about Venus's physical diameter now allows the relative geometry to be converted into actual distances.
Using the largest observed angular diameter:
The nearest Earth–Venus distance is therefore
At inferior conjunction,
and since
we obtain
which gives
or approximately
Given the observational limitations of the period, an uncertainty of perhaps 20–30% would be entirely plausible:
The modern value is
which falls comfortably within this range.
Step 5: Estimating Earth's Orbital Speed
Once the size of Earth's orbit is known, its orbital speed follows immediately from its period of one year.
Using
gives
Allowing for the uncertainty in the AU estimate, the likely range would be roughly 25–40 km/s.
The modern value is approximately 29.8 km/s.
Step 6: Estimating the Sun's Diameter
The Sun's apparent angular diameter can be measured directly and is approximately
Converting to radians:
The Sun's diameter is therefore
Given the uncertainty in the AU estimate, the inferred solar diameter would also carry an uncertainty of roughly 20–30%.
The modern value is
again well within the expected range.
Sources of Error
- Venus is assumed to have the same diameter as Earth.
- The orbits are treated as circular.
- Orbital inclinations are neglected.
- The small-angle approximation is used.
- Planetary diameters are difficult to measure accurately with seventeenth-century telescopes.
- Venus's changing phase complicates diameter measurements.
- Atmospheric turbulence and imperfect optics introduce additional uncertainty.
These effects make percent-level precision unrealistic. A seventeenth-century astronomer would be justified in claiming only the approximate scale of the Solar System, not exact values.
Conclusion
The remarkable fact is that most of the geometry of the Solar System was already known by the time of Newton. What was missing was an absolute length scale.
By assuming that Venus is approximately Earth-sized and measuring its apparent diameter, an astronomer could plausibly estimate the Earth–Sun distance, Earth's orbital speed, and the Sun's physical diameter to within a few tens of percent.
The important achievement would not be numerical precision but the realization that the Solar System is vastly larger than everyday terrestrial distances and that planetary motions occur on scales of hundreds of millions of kilometers.
Appendix: Estimating the Speed of Light Using Rømer's Observations
Having estimated the astronomical unit, we can use another famous seventeenth-century observation to estimate the speed of light.
In 1676, Ole Rømer noticed that the eclipses of Jupiter's moon Io appeared earlier or later depending on the position of Earth in its orbit.
As Earth moves from one side of its orbit to the other, the distance traveled by light changes by approximately
Using our estimate,
Rømer's observations suggested a cumulative delay on the order of twenty minutes. Taking
gives
The modern value is
Given the uncertainty in both the AU estimate and the eclipse-delay measurements, the calculation should be regarded as an order-of-magnitude determination rather than a precise measurement. Nevertheless, it correctly places the speed of light in the range of several hundred thousand kilometers per second.
Most importantly, the calculation correctly establishes that light travels at a finite speed of order \(10^8\) meters per second, an extraordinary conclusion for seventeenth-century astronomy.
Why Assume Venus Is Approximately Earth-Sized?
The assumption that Venus is approximately the same size as Earth may initially appear arbitrary. However, a seventeenth-century astronomer would not have been making this choice without observational guidance.
Although the absolute sizes of the planets were not yet known, several independent lines of evidence provided indirect constraints on their relative scales.
Earth already provided a known planetary diameter from geodetic measurements, and the Moon’s size had been estimated from its distance and angular diameter:
This established that celestial bodies could plausibly range from a few thousand kilometers to Earth-sized scales.
Telescopic observations showed that Venus, Mars, Jupiter, and Saturn all appeared as resolved disks rather than point-like stars. When combined with Keplerian relative distances, this allowed astronomers to estimate ratios of planetary diameters even without knowing the absolute scale of the Solar System.
These comparisons suggested that Venus was substantially larger than the Moon and broadly comparable to the other major planets. It did not appear to belong to a fundamentally different class of object. In particular, Venus occupies an orbit adjacent to Earth’s and exhibits phases similar to those of the Moon, reinforcing the interpretation of Venus as a “world” of Earth-like character.
Additional constraints arise from transits of Venus across the Sun and lunar occultations of Venus. These events allow Venus’s angular size to be compared against well-defined celestial references, providing more reliable estimates than isolated telescopic disk measurements.
None of these observations uniquely determine the physical diameter of Venus. Any assumed value can be rescaled consistently by adjusting the size of the entire Solar System. However, not all scalings are equally plausible.
A Venus significantly smaller than the Moon would imply that a prominent planetary body is physically minor compared to Earth’s satellite, despite its strong telescopic presence and planetary behavior. Conversely, a Venus several times larger than Earth would imply a correspondingly enlarged Solar System, a much larger Sun, and higher inferred orbital speeds, producing a far more extreme cosmic scale.
A cautious astronomer seeking the simplest model consistent with all available evidence might therefore prefer a Venus of the same order of magnitude as Earth. A plausible range would be:
with an Earth-sized Venus representing a natural central estimate.
As it happens, the modern value is:
only about five percent smaller than Earth’s diameter.
The assumption adopted in this essay is therefore not merely convenient; it is consistent with a broad network of indirect observational constraints available to seventeenth-century astronomy.
