The Vicarious Hypothesis, or ''hypothesis vicaria'', was a planetary

hypothesis A hypothesis (: hypotheses) is a proposed explanation for a phenomenon. A scientific hypothesis must be based on observations and make a testable and reproducible prediction about reality, in a process beginning with an educated guess o ...

proposed by

Johannes Kepler Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...

to describe the

motion In physics, motion is when an object changes its position with respect to a reference point in a given time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and frame of reference to an o ...

Mars Mars is the fourth planet from the Sun. It is also known as the "Red Planet", because of its orange-red appearance. Mars is a desert-like rocky planet with a tenuous carbon dioxide () atmosphere. At the average surface level the atmosph ...

. The hypothesis adopted the

circular orbit A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle. In this case, not only the distance, but also the speed, angular speed, Potential energy, potential and kinetic energy are constant. T ...

and

equant Equant (or punctum aequans) is a mathematical concept developed by Claudius Ptolemy in the 2nd century AD to account for the observed motion of the planets. The equant is used to explain the observed speed change in different stages of the plane ...

of Ptolemy's planetary model as well as the

heliocentrism Heliocentrism (also known as the heliocentric model) is a superseded astronomical model in which the Earth and planets orbit around the Sun at the center of the universe. Historically, heliocentrism was opposed to geocentrism, which placed t ...

of the

Copernican model Copernican heliocentrism is the astronomical model developed by Nicolaus Copernicus and published in 1543. This model positioned the Sun at the center of the Universe, motionless, with Earth and the other planets orbiting around it in circular pa ...

. Calculations using the Vicarious Hypothesis did not support a circular orbit for Mars, leading Kepler to propose elliptical orbits as one of three laws of planetary motion in '' Astronomia Nova''.

History

In 1600, Johannes Kepler met and began working with

Tycho Brahe Tycho Brahe ( ; ; born Tyge Ottesen Brahe, ; 14 December 154624 October 1601), generally called Tycho for short, was a Danish astronomer of the Renaissance, known for his comprehensive and unprecedentedly accurate astronomical observations. He ...

at Benátky, a town north of

Prague Prague ( ; ) is the capital and List of cities and towns in the Czech Republic, largest city of the Czech Republic and the historical capital of Bohemia. Prague, located on the Vltava River, has a population of about 1.4 million, while its P ...

where Brahe's new observatory was being built. Brahe assigned Kepler the task of modeling the motion of Mars using only data that Brahe had collected himself. Upon the death of Brahe in 1601, all of Brahe's data was willed to Kepler. Brahe's observational data was among the most accurate of his time, which Kepler used in the construction of the Vicarious Hypothesis.

Predecessors

Ptolemy

Claudius Ptolemy's planetary model consisted of a stationary earth surrounded by fixed circles, called deferents, which carried smaller,

rotating Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersec ...

circles called epicycles. Planets rotated on the epicycles as the epicycles traveled along the deferent. Ptolemy shifted the Earth away from the center of the deferent and introduced another point, the

, equidistant to the deferent's center on the opposite side of the Earth. The Vicarious Hypothesis uses a circular orbit for Mars and reintroduces a form of the equant to describe the motion of Mars with constant

angular speed In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine f ...

Copernicus

Nicolaus Copernicus Nicolaus Copernicus (19 February 1473 – 24 May 1543) was a Renaissance polymath who formulated a mathematical model, model of Celestial spheres#Renaissance, the universe that placed heliocentrism, the Sun rather than Earth at its cen ...

broke from the

geocentric model In astronomy, the geocentric model (also known as geocentrism, often exemplified specifically by the Ptolemaic system) is a superseded scientific theories, superseded description of the Universe with Earth at the center. Under most geocentric m ...

of Ptolemy by placing the Sun at the center of his planetary model. However, Copernicus retained circular orbits for the planets and added an

orbit In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...

for the Earth, insisting that the Earth revolved around the Sun. The Sun was positioned off-center of the orbits but was still contained within all orbits. Kepler adopted Copernican heliocentrism in the construction of the Vicarious Hypothesis so that his measurements of the distances to Mars were taken relative to the Sun.

Development

Kepler's construction of the Vicarious Hypothesis was based on a circular orbit for Mars and a heliocentric model for the planets. After receiving

longitudinal Longitudinal is a geometric term of location which may refer to: * Longitude ** Line of longitude, also called a meridian * Longitudinal engine, an internal combustion engine in which the crankshaft is oriented along the long axis of the vehicle, ...

observation data from Tycho Brahe, Kepler had twelve observations, two being his own, in which Mars was at

opposition Opposition may refer to: Arts and media * ''Opposition'' (Altars EP), 2011 EP by Christian metalcore band Altars * The Opposition (band), a London post-punk band * ''The Opposition with Jordan Klepper'', a late-night television series on Comedy ...

to the Sun. From these twelve observations, Kepler chose four to form the basis of the Vicarious Hypothesis because they had a relatively uniform distribution across his proposed circular orbit for Mars. In this sense, the Vicarious Hypothesis functions as a fit to observational data.{{Cite journal, last=Whiteside, first=D. T., date=1974-02-01, title=Internationales Kepler-Symposium Weil der Stadt 1971. Referate und diskussionen: F. Krafft, K. Meyer and B. Sticker, ed. H.A. Gerstenberg: Hildesheim, 1973. xii + 490 pp. DM 160, journal=Studies in History and Philosophy of Science Part A, volume=4, issue=4, pages=387–392, doi=10.1016/0039-3681(74)90010-7, bibcode=1974SHPSA...4..387W, issn=0039-3681 Kepler used these four observations to determine the eccentricities of the Sun and equant of his proposed orbit. Unlike the Ptolemaic System, in which the Earth and equant were assumed equidistant to the center of the orbit, the Vicarious Hypothesis placed the equant where the time and location of the observation would match. Using the Vicarious Hypothesis, Kepler determined the eccentricities of the Sun and equant to be 11,332 and 7,232 arbitrary units, respectively, for the Martian orbital

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

of 100,000 units. Using these positions for the Sun and equant, the model constructed using the Vicarious Hypothesis agreed with the twelve observations within 2' of arc, a level of accuracy better than any other previous model. While the heliocentric longitudes of this model proved to be accurate, distances from the Sun to Mars, or

latitudes In geography, latitude is a geographic coordinate that specifies the north-south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from −90° at the south pole to 90° at the ...

of Mars, challenged the model. In his book, ''Astronomia Nova'', Kepler determined that the eccentricity of the Sun, based on latitudinal oppositions, should be between a range of 8,000 and 9,943, conflicting with the eccentricity of 11,332 determined by the Vicarious Hypothesis. To accommodate the latitudinal data, Kepler modified the Vicarious Hypothesis to include a bisected eccentricity, making the Sun and equant equidistant to the center of the orbit. This resolved the error in the latitudes of Mars but introduced a longitudinal error of 8' of arc in some parts of the Mars orbit. While an 8' error still had better accuracy than previous models, corresponding to approximately one-fourth the

diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

of the

Moon The Moon is Earth's only natural satellite. It Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...

, Kepler rejected the Vicarious Hypothesis because he did not believe it was accurate enough to model the true orbit of Mars.

Historical significance

The errors in latitude and longitude of the Mars orbit made Kepler realize that false assumptions were made using the Vicarious Hypothesis. In particular, Kepler amended the hypothesis to exclude the circular orbit. Kepler realized that he could fix the error by reducing the spread of the central region of the circular orbit, creating an

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

. He used calculations previously made with the Vicarious Hypothesis to confirm the elliptical orbit for Mars. Kepler published his results in ''Astronomia Nova'', in which he introduces the elliptical orbit for planets as his first law of planetary motion.

References

Astronomy Johannes Kepler Mars Planetary habitability Space colonization