Gravitational dances: Why planets don’t collide or pull each other off course in the solar system

Both planets and moons move around the Sun, which is the central, most massive, most influential and attractive body in the Solar system. The moons or natural satellites are kept in orbit by the gravitational attraction between the satellite and the celestial body or planet it orbits.

The Earth has a much greater mass than the Moon. The Moon orbits planet Earth due to the gravitational pull of Earth. The Earth moves in an elliptical orbit around the Sun due to the gravitational attraction of the Sun. While the Moon is orbiting the Earth, both the Moon and Earth move around the Sun. The same applies for artificial satellites, and for other planets and their moons or natural satellites.

The question of the stability of planetary orbits, or why planets “don’t pull each other off course”, is related to topics and disciplines such as the stability of the solar system, the n-body problem, gravitation, and celestial mechanics.

Gravity, or gravitational attraction, provides the force needed to maintain the stable orbit of two or more planets around a star and also of moons and artificial satellites around a planet.

Newton’s law of universal gravitation states that every point mass attracts every single other point mass by a force acting along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them.

This law can be expressed as:

Newton’s law applies to all celestial bodies. The planet Jupiter is attracted by the Sun, as is planet Saturn, and Jupiter and Saturn attract each other. The Sun’s influence is the most dominant one because the mass of the Sun represents approximately 99.8% of the mass of the solar system.

Within a planetary system, planets, dwarf planets, asteroids and other minor planets, comets, and space debris orbit the system’s barycenter in elliptical orbits.

The general problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally is called the n-body problem.

For mass m_i with position vector q_i, the n-body equations of motion (summing over all masses) can be written as:

The general n-body problem is difficult to solve analytically. It is mostly solved or simulated using numeral methods and power series solutions.

Investigating the stability of the solar system led scientists and mathematicians in the 18th century to create a new method of calculation in celestial mechanics known as the method of perturbations.

The orbits and motions of the planets Jupiter and Saturn, mentioned in this question, have a significance in the history of astronomy, and have been specifically studied and explained by Laplace at the end of the 18th century.

Laplace presented a memoir on planetary inequalities in three sections, in 1784, 1785, and 1786. This dealt with the identification and explanation of the perturbations now known as the “great Jupiter–Saturn inequality”. Laplace solved a longstanding problem in the study and prediction of the movements of these planets, showing by general considerations, that the mutual action of two planets could never cause large changes in the eccentricities and inclinations of their orbits; then, that peculiarities arose in the Jupiter–Saturn system because of the near approach to commensurability of the mean motions of Jupiter and Saturn.

The two planets’ mean motions is very nearly equal to a ratio between a pair of small whole numbers. Two periods of Saturn’s orbit around the Sun almost equal five of Jupiter’s. The corresponding difference between multiples of the mean motions, (2n_J−5n_S), corresponds to a period of nearly 900 years, and it occurs as a small divisor in the integration of a very small perturbing force with this same period. As a result, the integrated perturbations with this period are disproportionately large, about 0.8° degrees of arc in orbital longitude for Saturn and about 0.3° for Jupiter. With the help of Laplace’s discoveries, the tables of the motions of Jupiter and Saturn could at last be made much more accurate.

Moreover, Laplace was able to explain Claudius Ptolemy’s astronomical observations in Antiquity to about one minute of arc, with no need for additional terms in the calculations. Hence he demonstrated that Newton’s law of universal gravitation was enough to explain the motion of the planets all over known history.

Laplace and Lagrange showed that the semi-major axes of the planets undergo small oscillations, and do not have secular terms, which represented the first significant result in explaining the Stability of the Solar System.

The next generations of humans will not need to worry about planets colliding into each other or pulling each other off course.

Computer simulations and studies have shown that in the next few hundreds of millions of years the motions and orbits of the planets will mostly remain regular and stable, with the possibility of appearance of chaotic orbits after tens of millions of years.
It is to be noted that in less than five billion years it’s possible that collisions between planets might take place, or also ejections of planets out of the solar system.

The importance of using mathematics in the sciences, and some cases where math is not needed in physics

Simply put, without mathematics there is no exact science: no physics, no chemistry, no (scientific and accurate ) biology or medicine …
Mathematics is an essential component of the scientific method; data collection and observation and hypotheses are sustained by relevant mathematical models, and mathematics is the language in which the Natural or Physical World is written, as Galileo once said.
Can anyone imagine physics without numbers , or without derivatives, integrals, equations , tensors, geometrical figures, etc ? Physics would be a barren field of study based on verbal statements with no precision.
The xkcd image below expresses or conveys the importance of mathematics:

Math & the sciences

All the exact sciences are useful and important, but they all need and use mathematics.
That being said, there are a few cases or circumstances where (as an exception) mathematics may temporarily not be needed in theoretical physics.
When a teacher is explaining general concepts in (theoretical) physics to students who are beginning to learn physics, he may not use mathematics.
When a scientist or physicist is giving a lecture to a general audience, he may qualitatively describe physical theories and phenomena without using mathematics.
Similarly, when a scientist writes a popularization book about physics, he/she may use words (and some images) to present or explain concepts and topics in physics, without mathematics.
When one is imagining or mentally elaborating a thought experiment, considering some hypothesis, theory, or principle for the purpose of thinking through its consequences and repercussions, mathematics is generally not used.
When observing a physical phenomenon (with the naked eye or without instruments) for the first time, or when conducting general qualitative observations, mathematics may not be used, but math will be used afterwards to collect, interpret and classify data and to formulate or build a coherent theory.
The above were particular instances of mathematical equations, formulas and tools temporarily not being used or needed in (theoretical) physics.

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Here are some additional thoughts or remarks about math, physics, and the people who work or “dabble” with these topics or disciplines.

A mathematical theory that is coherent, logical and self-consistent can be regarded as beautiful. Mathematical shapes, curves, solids and surfaces can also be described as “beautiful”. However consistency, coherence and beauty (or elegance) are not enough in relation to physics.

Physical theories must be coherent and follow the rules of the scientific method, which means they use correct and adequate mathematical tools and models, but they also rely on observations, on scientific data, and they must be testable and experimentally verified.

Conceptions, criteria and/or perceptions of beauty or of what is considered beautiful may change with time with regard to science. Some perceptions might be personal, subjective or philosophical.

The tendency in physics during recent years or during the last few decades to formulate sophisticated theories that are regarded as “beautiful” but are difficult to test or verify has sometimes lead to stagnation and to a counterproductive lack of progress.

Interdisciplinary or multidisciplinary knowledge is always possible and helpful. A physicist could find or elaborate the necessary or appropriate math to be used in a physical theory or in theoretical physics, but this depends on the prior knowledge and skills the physicist has in relation to both math and physics.
Moreover, it would be beneficial if people who are known as mathematicians do not get haughty or boastful with others, and the educational methods, frameworks and systems within which mathematics and the sciences are taught ought to be reassessed regularly and reformed when or if necessary.