This article relies too much on references to primary sources . (November 2008) (Learn how and when to remove this template message)
On the Sizes and Distances (of the Sun and Moon) (Περὶ μεγεθῶν καὶ ἀποστημάτων [ἡλίου καὶ σελήνης], Peri megethon kai apostematon) is widely accepted as the only extant work written by Aristarchus of Samos, an ancient Greek astronomer who lived circa 310–230 BCE. This work calculates the sizes of the Sun and Moon, as well as their distances from the Earth in terms of Earth's radius.
The book was presumably preserved by students of Pappus of Alexandria's course in mathematics, although there is no evidence of this. The editio princeps was published by John Wallis in 1688, using several medieval manuscripts compiled by Sir Henry Savile.The earliest Latin translation was made by Giorgio Valla in 1488. There is also a 1572 Latin translation and commentary by Frederico Commandino.
The work's method relied on several observations:
The rest of the article details a reconstruction of Aristarchus' method and results.The reconstruction uses the following variables:
|φ||Angle between the Moon and the Sun during a half moon (directly measurable)|
|L||Distance from the Earth to the Moon|
|S||Distance from the Earth to the Sun|
|ℓ||Radius of the Moon|
|s||Radius of the Sun|
|t||Radius of the Earth|
|D||Distance from the center of Earth to the vertex of Earth's shadow cone|
|d||Radius of the Earth's shadow at the location of the Moon|
|n||Ratio, d/ℓ (a directly observable quantity during a lunar eclipse)|
|x||Ratio, S/L = s/ℓ (which is calculated from φ)|
Aristarchus began with the premise that, during a half moon, the moon forms a right triangle with the Sun and Earth. By observing the angle between the Sun and Moon, φ, the ratio of the distances to the Sun and Moon could be deduced using a form of trigonometry.
From the diagram and trigonometry, we can calculate that
The diagram is greatly exaggerated, because in reality, S = 390 L, and φ is extremely close to 90°. Aristarchus determined φ to be a thirtieth of a quadrant (in modern terms, 3°) less than a right angle: in current terminology, 87°. Trigonometric functions had not yet been invented, but using geometrical analysis in the style of Euclid, Aristarchus determined that
In other words, the distance to the Sun was somewhere between 18 and 20 times greater than the distance to the Moon. This value (or values close to it) was accepted by astronomers for the next two thousand years, until the invention of the telescope permitted a more precise estimate of solar parallax.
Aristarchus also reasoned that as the angular size of the Sun and the Moon were the same, but the distance to the Sun was between 18 and 20 times further than the Moon, the Sun must therefore be 18–20 times larger.
Aristarchus then used another construction based on a lunar eclipse:
By similarity of the triangles, and
Dividing these two equations and using the observation that the apparent sizes of the Sun and Moon are the same, , yields
The rightmost equation can either be solved for ℓ/t
The appearance of these equations can be simplified using n = d/ℓ and x = s/ℓ.
The above equations give the radii of the Moon and Sun entirely in terms of observable quantities.
The following formulae give the distances to the Sun and Moon in terrestrial units:
where θ is the apparent radius of the Moon and Sun measured in degrees.
It is unlikely that Aristarchus used these exact formulae, yet these formulae are likely a good approximation for those of Aristarchus.
The above formulae can be used to reconstruct the results of Aristarchus. The following table shows the results of a long-standing (but dubious) reconstruction using n = 2, x = 19.1 (φ = 87°) and θ = 1°, alongside the modern day accepted values.
|s/t||Sun's radius in Earth radii||6.7||109|
|t/ℓ||Earth's radius in Moon radii||2.85||3.50|
|L/t||Earth-Moon distance in Earth radii||20||60.32|
|S/t||Earth-Sun distance in Earth radii||380||23,500|
[ citation needed ]
The error in this calculation comes primarily from the poor values for x and θ. The poor value for θ is especially surprising, since Archimedes writes that Aristarchus was the first to determine that the Sun and Moon had an apparent diameter of half a degree. This would give a value of θ = 0.25, and a corresponding distance to the Moon of 80 Earth radii, a much better estimate. The disagreement of the work with Archimedes seems to be due to its taking an Aristarchus statement that the lunisolar diameter is 1/15 of a "meros" of the zodiac to mean 1/15 of a zodiacal sign (30°), unaware that the Greek word "meros" meant either "portion" or 7°1/2; and 1/15 of the latter amount is 1°/2, in agreement with Archimedes' testimony.
A similar procedure was later used by Hipparchus, who estimated the mean distance to the Moon as 67 Earth radii, and Ptolemy, who took 59 Earth radii for this value.
Some interactive illustrations of the propositions in On Sizes can be found here:
In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation intersects a localized phenomenon. For example, the Rutherford cross-section is a measure of probability that an alpha-particle will be deflected by a given angle during a collision with an atomic nucleus. Cross section is typically denoted σ (sigma) and is expressed in units of transverse area. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen constitutes about 75% of the baryonic mass of the universe.
In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The three laws state that:
Tidal acceleration is an effect of the tidal forces between an orbiting natural satellite, and the primary planet that it orbits. The acceleration causes a gradual recession of a satellite in a prograde orbit away from the primary, and a corresponding slowdown of the primary's rotation. The process eventually leads to tidal locking, usually of the smaller first, and later the larger body. The Earth–Moon system is the best-studied case.
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as
In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space.
Aristarchus of Samos was an ancient Greek astronomer and mathematician who presented the first known heliocentric model that placed the Sun at the center of the known universe with the Earth revolving around it. He was influenced by Philolaus of Croton, but Aristarchus identified the "central fire" with the Sun, and he put the other planets in their correct order of distance around the Sun. Like Anaxagoras before him, he suspected that the stars were just other bodies like the Sun, albeit farther away from Earth. His astronomical ideas were often rejected in favor of the geocentric theories of Aristotle and Ptolemy. Nicolaus Copernicus attributed the heliocentric theory to Aristarchus.
In geometry, a solid angle is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle from that point.
Earth radius is the distance from the center of Earth to a point on its surface. Its value ranges from a near maximum 6,378 km (3,963 mi) at the equator to a near minimum 6,357 km (3,950 mi) at either pole. A nominal Earth radius is sometimes used as a unit of measurement in astronomy and geophysics, denoted in astronomy by the symbol R⊕. In other contexts, it is denoted or sometimes . The early definition of the metre such that the distance from equator to pole along the circumference is 10,000 km gives a radius roughly 6,367 km which is close to halfway between the minimum and maximum. However a better “average” is usually considered to be 6,371 km with a 0.3% variability for the following reasons.
A tautochrone or isochrone curve is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point on the curve. The curve is a cycloid, and the time is equal to π times the square root of the radius over the acceleration of gravity. The tautochrone curve is related to the brachistochrone curve, which is also a cycloid.
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
In Einstein's theory of general relativity, the Schwarzschild metric is the solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun. It was found by Karl Schwarzschild in 1916, and around the same time independently by Johannes Droste, who published his much more complete and modern-looking discussion only four months after Schwarzschild.
The Hill sphere or Roche sphere of an astronomical body is the region in which it dominates the attraction of satellites. The outer shell of that region constitutes a zero-velocity surface. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sphere of its own. Any object within that distance would tend to become a satellite of the moon, rather than of the planet itself. One simple view of the extent of the Solar System is the Hill sphere of the Sun with respect to local stars and the galactic nucleus.
Spacecraft flight dynamics is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag.
In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy.
A photon sphere or photon circle is an area or region of space where gravity is so strong that photons are forced to travel in orbits. The radius of the photon sphere, which is also the lower bound for any stable orbit, is, for a Schwarzschild black hole:
On Sizes and Distances is a text by the ancient Greek astronomer Hipparchus in which approximations are made for the radii of the Sun and the Moon as well as their distances from the Earth. It is not extant, but some of its contents have been preserved in the works of Ptolemy and his commentator Pappus of Alexandria. Several modern historians have attempted to reconstruct the methods of Hipparchus using the available texts.
In classical mechanics, Newton's theorem of revolving orbits identifies the type of central force needed to multiply the angular speed of a particle by a factor k without affecting its radial motion. Newton applied his theorem to understanding the overall rotation of orbits that is observed for the Moon and planets. The term "radial motion" signifies the motion towards or away from the center of force, whereas the angular motion is perpendicular to the radial motion.
Geographical distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. This distance is an element in solving the second (inverse) geodetic problem.
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the widest points of the perimeter.