How the Earth-Sun distance was measured for the first time

Relative size of the solar system

From relative distances to absolute scale

At the beginning of the 17th century, scientists had no idea of the dimensions of the solar system. Thanks to the laws of Johannes Kepler (1571−1630), the relative distances of the five planets known at the time with respect to the Sun were known. In other words, the distances were known in an unknown unit, the Earth-Sun distance, but the Earth-Sun distance in km was not known.

Kepler's third law

The 3rd law of Johannes Kepler, formulated in 1618, shows the relationship between the semi-major axis of a planet's orbit and its period of revolution around the Sun (table opposite). The cube of the semi-major axis is equal to the square of the period of revolution (T2 / R3 = k), which places Mercury's orbit at 0.39 AU (astronomical unit created much later in 1958) and Saturn's at 9.55 astronomical units from the Sun.

N.B.:
Kepler's first law "law of orbits" (published in 1609 in Astronomia nova), states that all orbiting bodies describe elliptical trajectories, with the Sun occupying one of the foci.
Kepler's second law "law of areas" (published in 1609 in Astronomia nova), states that equal areas are swept out in equal times.
Kepler's third law "law of periods" (published in 1619 in Harmonices Mundi), states that the square of the sidereal period of a planet (time between two successive passages in front of a star) is directly proportional to the cube of the semi-major axis of the planet's elliptical trajectory.

Orbital characteristics of the planets

The table below presents the orbital data of the planets of the solar system, illustrating Kepler's third law.

N.B.:
T² / R³ = k: the theoretical expression of k would become known in 1687 thanks to the equations of universal gravitation formulated by Isaac Newton (1643−1727).

A single measurement for the whole system

The measurement of a single distance (planet-Sun) would give the scale of the entire system and therefore the long-awaited Earth-Sun distance in km. In 1687, Isaac Newton discovered the law of gravitation, which allowed him to explain Kepler's three laws.

Measurement of the Earth-Mars distance

The principle of parallax

However paradoxical it may seem, it was by measuring the Earth-Mars distance that the astronomers and mathematicians of the 17th century were able to gain an idea of the dimensions of the solar system. At that time, observation consisted mainly of measuring the angles of celestial objects seen from the Earth. Then trigonometric calculations gave the distances.

How was the Earth-Mars distance calculated for the first time?

The simultaneous Paris-Cayenne observation

In September 1672, Jean-Dominique Cassini (1625−1712), Jean Picard known as Abbé Picard (1620−1682) and Jean Richer (1630−1696) measured the horizontal parallax of Mars when the planet passed closest to the Earth (Mars is then, as seen from the Earth, on the opposite side of the Sun). To carry out this measurement, the positions of Mars relative to much more distant stars must be observed from two very distant points.

Cassini from Paris and Richer from Cayenne measured the parallax of Mars. This measurement, related to the base formed by the equatorial radius of the Earth, gives a horizontal parallax of p=24" ⇒ Earth-Mars = 54,746,000 km. To determine the parallax of Mars from observations made from Paris and Cayenne, it is necessary to know with maximum precision the difference between the latitudes and the difference between the longitudes of Paris and Cayenne.

Determination of longitudes

Latitude is easy to measure, but longitude is much more difficult at the time.

Cassini nevertheless specifies that he measured the longitude by several methods to obtain an average:

• Observation of an eclipse whose different phases are noted in hours of the Paris meridian, in astronomical compilations.
• Observation of eclipses of Jupiter's Galilean satellites, which occur several times a day.
• Observation of the meridian altitudes of the Sun.

The whole gives an average of 3 h 39 min ±10 min. Richer and Cassini therefore observe the planet Mars during the month of September 1672. Mars will pass near a star of Aquarius (Ψ Aquarii). In the image, point C' corresponds to the latitude of C (Cayenne) and the longitude of P (Paris). Cassini uses two simultaneous observations of Mars, one from P, the other from C'. The angle Paris Cayenne from Mars is the angle α=14". But the parallax of Mars is the angle under which Mars sees the radius of the Earth. The calculation of proportionality between the angle Paris Cayenne and the parallax of Mars gives 24" ± 5".

Calculation of the Earth-Mars distance

Once the parallax of Mars is known, trigonometry gives the Earth-Mars distance (D) as a function of the radius (R) of the Earth:
D = R×3600/24×180/π i.e. 8600 R
D = 8600 * 6371 = 54,790,600 km

N.B.:
Latitude is the north or south positioning of a point on Earth; in other words, it is the angle with the equatorial plane (from 0° at the equator to 90° at the north pole and -90° at the south pole).
Longitude is the east-west positioning of a point on Earth, which corresponds to a time offset with the plane of the Greenwich meridian (prime meridian, longitude 0°). This offset varies from +180° to the west to -180° to the east.

Calculation of the Earth-Sun distance

From the Earth-Mars distance to the astronomical unit

As soon as the Earth-Mars distance was known, the Earth-Sun distance could be calculated thanks to Kepler's third law.

Cassini knew that Mars was at \(0{,}38\ \text{AU} = \frac{3}{8}\ \text{AU}\). \[ \text{Earth-Sun} = \frac{8}{3} \times 8600\,R = 23\,000\,R \pm 5\,000\ \text{Earth radii}. \] Cassini gives us the value of the Earth's radius in leagues: \(R = 1500\) leagues.

Application of Kepler's third law

At the time, there were many definitions of the league depending on the region. Let us assume that the Earth's radius is given in Abbé Picard's league, which, from 1671, was equal to 3707 metres. \[ R = 1500 \times 3{,}707 = 5\,560{,}5\ \text{km} \] hence: \[ \text{Earth-Sun} = 127\,891\,500\ \text{km} \pm 27\,802\,500\ \text{km}. \]

This remarkable result constitutes the first measurement of the Earth-Sun distance. Thanks to Kepler's third law, this distance would give the scale of the entire solar system \(T^2/R^3 = k\).

Calculation of distances from the orbits of the Earth and Mars

Geometric relationships between the orbits

\(C =\) Centre of the ellipse representing the orbit of Mars
\(CM_{\text{mars}} =\) Semi-major axis of the orbit of Mars
Eccentricity of the orbit of Mars \(= \dfrac{CS_{\text{sun}}}{CM_{\text{mars}}} = 0{,}093\)
\(CM_{\text{mars}} = CS_{\text{sun}} + S_{\text{sun}}T_{\text{earth}} + T_{\text{earth}}M_{\text{mars}}\)
\(CM_{\text{mars}} = T_{\text{earth}}M_{\text{mars}} + \dfrac{S_{\text{sun}}T_{\text{earth}}}{1-CS_{\text{sun}}}\)
\(T_{\text{earth}}M_{\text{mars}} = 54\,790\,600\ \text{km}\)
\(PT_{\text{earth}}\) (period of the Earth) \(= 1\ \text{year} (365\ \text{days})\)
\(PM_{\text{mars}}\) (period of Mars) \(= 1{,}88\ \text{year} (686\ \text{days})\)

Application of Kepler's third law

\[ T_{\text{earth}}S_{\text{sun}} = \frac{T_{\text{earth}}M_{\text{mars}}}{1 - CS_{\text{sun}}\left(\frac{PM_{\text{mars}}}{PT_{\text{earth}}}\right)^{2/3} - 1} \]

Result of the calculation

\[ 1 - CS_{\text{sun}}\left(\frac{PM_{\text{mars}}}{PT_{\text{earth}}}\right)^{2/3} - 1 = 0{,}38 \quad \text{i.e.} \quad \frac{3}{8} \] \[ \Rightarrow T_{\text{earth}}S_{\text{sun}} = 54\,790\,600 \times \frac{8}{3} = 150\,000\,000\ \text{km} \]