Math in Astronomy 101

math in astronomy

Math or Mathematics and astronomy have a long history.

The two sciences were once inseparable and remain so today. While physics provides much of the mathematics used to interpret the knowledge gained from astronomical observations, some special needs conditions intertwine mathematics with phenomena that do not yet have adequate physics to describe the observations.

Both of these uses of mathematics make mathematical astronomy an ongoing challenge.

What type of math do astronomers use?

Unlike “classical” astronomy, astronomers conduct little or no research. His primary responsibility is to develop mathematical models that are as precise as possible. Astronomer explains the formation of the celestial bodies, as well as their future and history.

It is involved in moving the planets and satellites of our solar system, as well as :

  •  -exoplanets
  •  -small bodies (from dwarf planets to dust, via asteroids and comets), — – artificial satellites
  • – their debris using complex equations that combine their evolution with their spinning, their past with their inner structure.

One of astronomers considerations is the stability of these orbits, as well as many of the powers that may add to gravity, such as : 

  • stresses
  • relativistic impacts
  • gas phenomena
  • and atmosphere brakes

  …. which greatly complicates the explanation of the structures. 

Technology and mathematics that has been used in space exploration missions and has led to significant discoveries.

The Neptune planet was discovered by mathematical computation and not by empirical experimentation, as some did.

Neptun
The Neptune planet (Crédits : NASA)

This profession is often compared to that of a cosmologist, who is more interested in the universe as a whole, its planetary history, rather than human bodies, and is more commonly a teaching physicist.

Strong computers and specialized software are needed to complete all of this work. These mathematicians are also called upon to present lectures or to publish papers in specialized journals. 

Math that astronomers are using:

  • Trigonometry
  • Conics
  • Differential and integral calculus
  • Differential geometry
  • Statistics and probabilities

How astronomers use algebra?

Almost all astronomers are scholars, which necessitates the use of algebra. Some are scientists who are focusing on rules governing the formation of astronomical phenomena. Others examine massive volumes of data gathered from observatories and satellites and write papers about their observations. Certain astronomers usually work as part of a team to run ground-based telescopes.

However, astronomers may spend only a few weeks each year making observations with optical telescopes, radio telescopes, and other instruments. Over the years, satellites and other space-based instruments have provided us with astronomical data. 

New technologies that improve analytical techniques and instruments, such as computers, telescopes, and mounts, are reappearing in ground-based research. In museums that house planetariums, a limited number of astronomers operate. These astronomers create programs that are publicly viewed and will guide the planetarium’s operations.

hubble telescope
Photo of the Hubble space telescope during mission STS-125 (2009). (Crédits NASA / Wikipedia) 

Often astronomers use algebra in their work to achieve their goals. Without algebra, it will be difficult to find such things as the Hubble space telescope (HST). Algebra is used to measure speed and monitor motion rate. There is that some astronomers best represented those achievements. Without them, their use of algebraic science and technology will not flourish. 

Our knowledge of stars and space is based on those astronomers. 

How astronomers use math in their jobs? 

Astronomers have used mathematicians in the past for complex calculations. This is particularly useful when using a telescope to observe celestial objects.

The telescope’s camera simply records a series of numbers (basically translates or measures photons or electrons and records a sequence of numbers), which may refer to the sum of light produced from various objects in the sky (stars, star clusters …), the type of light, etc. In order to understand the information contained in these numbers, we must use mathematics and statistics to explain them.

Astronomers also use mathematics to establish and test hypotheses regarding the physical laws that govern celestial phenomena. The principle is made up of formulas that compare quantities. (A basic example is Newton’s second law, which states that force equals mass times acceleration).

In order to test these theories and use them to predict what we will observe in the sky, astronomers must use mathematics to manipulate the equation.

Isasac Newton
Isaac Newton  (Crédit Wikipedia) 

What kind of math is used in astronomy?

Mathematics and astronomy, these two sciences have developed together and have enriched each other, hence in astronomy, precise kinds of mathematics are used to discover the maximum of the mysterious secrets of the sky.

Trigonometry

One of the first duties assigned to astronomy was to explain and chart the sky’s changes. To do so, they needed to create appropriate geometric instruments. Trigonometry was the primary instrument of ancient astronomy (the word comes from Greek and means “measurement of triangles”; it is, therefore, the art of measuring angles). Indeed, the need for such a science arose to measure the distance between two stars by measuring the angle they form between them from the observer’s point of view.

The Greek astronomer Hipparchus, who lived in the second century B.C, founded trigonometry and calculated the first trigonometric tables (calculation of sine, cosine, and tangent of an angle) in order to predict regular astronomical phenomena. He developed a method to measure the ratio of distances between the Earth, the Moon, and the Sun.

When the Moon is exactly at one of the quarters (i.e., when we see precisely half of it), ESM is a right-angled triangle in M. The angle Alpha≈ 89.52 °, or a distance Earth-Sun 390 times greater than the distance Earth-Moon; Aristarchus found Alpha≈ 87 ° and a distance Earth-Sun only 19 times greater than the distance Earth-Moon. The method of Aristarchus was, therefore, good, but the measurements not very accurate.

Observations allowed us to quantify the lengths of chords and angles; a mathematical method allowing us to move from a chord to the corresponding arc of a circle in order to determine the lengths of trajectory was thus required: the trigonometric calculation was born, thanks to Ptolemy who discovered these formulas.

Conics 

Conics are curves introduced in the 4th century B.C. by the Greek mathematician Menechma in order to solve the purely mathematical problem of the duplication of the cube. Another Greek, Pappus of Alexandria (4th century), described them as the intersection of a plane and a cone. 

Thus, these conic sections, thus defined by the Greeks, highlight different types of curves that can be classified according to four major families: the circle, the ellipse, the parabola, and the hyperbola.

The nondegenerate conic sections

The ellipse became one of the most studied figures in astronomy since Kepler discovered orbital motion. Both bodies that orbit the Sun, including planets, comets, asteroids, and so on, have an elliptical orbit. The ray-vector that connects the center of the Earth to the center of the Sun sweeps equal regions for equal lengths of time.

Consequently, planet P has a higher speed when it is close to the Sun and a lower rate when it is far from the Sun. The speed of the planet is thus maximum at the perihelion and minimum at the aphelion.

illustration of Kepler’s second law, known as the “law of areas.”

h  

 ttps://www.youtube.com/watch?v=Dvoe8Ib5D1o 

Differential and integral calculus

The bases of differential and integral calculus were discovered in the second half of the 17th century by Leibniz (1646-1716) and Newton (1643-1727). Newton’s understanding of these mathematical ideas allowed him to develop the laws of “classical” mechanics, including the three basic laws of physics and the law of universal gravitation

Through calculation, Newton found Kepler’s laws describing the motion of the planets around the Sun.

In the Newtonian theory, the force of gravitational attraction between two bodies is inversely proportional to the square of their distance: 

In the middle of the 19th century, the astronomer’s John Couch Adam and Urbain Le Verrier established the existence of the planet Neptune by means of calculations. Their calculations were based on the perturbations of the observed orbit of Uranus with respect to its orbit predicted by Newton’s laws. At the beginning of the 20th century, the same type of method was used to predict the existence of Pluto.

Differential geometry

It was not until 1905 and 1916 and the formulation of the theory of Relativity (special and then general) by Albert Einstein (1879-1955) that an explanation of these phenomena was found. 

The mathematical framework of the theory of Relativity is that of differential geometry, precisely Lorentzian varieties. 

When Einstein formulated his theory, differential geometry was about sixty years old. It was Bernhard Riemann (1826-1866), who is 1854, who laid the foundations.

Riemann’s motivations were then purely mathematical. Einstein was oriented towards Riemann’s work and non-Euclidean geometries by his mathematician friend Marcel Grossmann (1878-1936), who also introduced him to the tensor calculus, which is the primary language of the relativity theory. 

In this new theory, gravitation is not a force but the manifestation of the deformation of space (in fact of space-time) produced by the distribution of matter.

A massive object deforms space-time.

Statistics and probabilities

One topic fascinates astrophysicists and, to a lesser extent, most of humankind in the hunt for exoplanets: is there life on another planet? 

The Earth demonstrates that a form of life that depends primarily on carbon may occur whether the mechanisms that caused life to evolve on Earth may have happened on another planet.

The question of extra-terrestrial life is not posed in terms of possibility but its probability, its frequency of occurrence. This mathematical tool is used to extract information from large samples of measurements. 

The mathematical principles outlined here for exoplanets apply to other situations. Statistics inevitably follows from theoretical questions such as “how many planets are there in our galaxy?” or “what masses do they have?” or “what distances do they have from their host stars?”. These questions are crucial to understand the formation and evolution of planets and to help in the search for extra-terrestrial life.

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