Perturbation (astronomy)

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The perturbing forces of the Sun on the Moon at two places in its orbit. The blue arrows represent the direction and magnitude of the gravitational force on the Earth. Applying this to both the Earth's and the Moon's position does not disturb the positions relative to each other. When it is subtracted from the force on the Moon (black arrows), what is left is the perturbing force (red arrows) on the Moon relative to the Earth. Because the perturbing force is different in direction and magnitude on opposite sides of the orbit, it produces a change in the shape of the orbit.

In astronomy, perturbation is the complex motion of a massive body subject to forces other than the gravitational attraction of a single other massive body. The other forces can include a third (fourth, fifth, etc.) body, resistance, as from an atmosphere, and the off-center attraction of an oblate or otherwise misshapen body.

Introduction

The study of perturbations began with the first attempts to predict planetary motions in the sky. In ancient times the causes were a mystery. Newton, at the time he formulated his laws of motion and of gravitation, applied them to the first analysis of perturbations, recognizing the complex difficulties of their calculation. Many of the great mathematicians since then have given attention to the various problems involved; throughout the 18th and 19th centuries there was demand for accurate tables of the position of the Moon and planets for marine navigation.

The complex motions of gravitational perturbations can be broken down. The hypothetical motion that the body follows under the gravitational effect of one other body only is typically a conic section, and can be readily described with the methods of geometry. This is called a two-body problem, or an unperturbed Keplerian orbit. The differences between that and the actual motion of the body are perturbations due to the additional gravitational effects of the remaining body or bodies. If there is only one other significant body then the perturbed motion is a three-body problem; if there are multiple other bodies it is an n-body problem. A general analytical solution (a mathematical expression to predict the positions and motions at any future time) exists for the two-body problem; when more than two bodies are considered analytic solutions exist only for special cases. Even the two-body problem becomes insoluble if one of the bodies is irregular in shape.

Mercury's orbital longitude and latitude, as perturbed by Venus, Jupiter and all of the planets of the Solar System, at intervals of 2.5 days. Mercury would remain centered on the crosshairs if there were no perturbations.

Most systems that involve multiple gravitational attractions present one primary body which is dominant in its effects (for example, a star, in the case of the star and its planet, or a planet, in the case of the planet and its satellite). The gravitational effects of the other bodies can be treated as perturbations of the hypothetical unperturbed motion of the planet or satellite around its primary body.

Mathematical analysis

General perturbations

In methods of general perturbations, general differential equations, either of motion or of change in the orbital elements, are solved analytically, usually by series expansions. The result is usually expressed in terms of algebraic and trigonometric functions of the orbital elements of the body in question and the perturbing bodies. This can be applied generally to many different sets of conditions, and is not specific to any particular set of gravitating objects. Historically, general perturbations were investigated first. The classical methods are known as variation of the elements, variation of parameters or variation of the constants of integration. In these methods, it is considered that the body is always moving in a conic section, however the conic section is constantly changing due to the perturbations. If all perturbations were to cease at any particular instant, the body would continue in this (now unchanging) conic section indefinitely; this conic is known as the osculating orbit and its orbital elements at any particular time are what are sought by the methods of general perturbations.

General perturbations takes advantage of the fact that in many problems of celestial mechanics, the two-body orbit changes rather slowly due to the perturbations; the two-body orbit is a good first approximation. General perturbations is applicable only if the perturbing forces are about one order of magnitude smaller, or less, than the gravitational force of the primary body. In the Solar System, this is usually the case; Jupiter, the second largest body, has a mass of about 1/1000 that of the Sun.

General perturbation methods are preferred for some types of problems, as the source of certain observed motions are readily found. This is not necessarily so for special perturbations; the motions would be predicted with similar accuracy, but no information on the configurations of the perturbing bodies (for instance, an orbital resonance) which caused them would be available.

Special perturbations

In methods of special perturbations, numerical datasets, representing values for the positions, velocities and accelerative forces on the bodies of interest, are made the basis of numerical integration of the differential equations of motion. In effect, the positions and velocities are perturbed directly, and no attempt is made to calculate the curves of the orbits or the orbital elements.

Special perturbations can be applied to any problem in celestial mechanics, as it is not limited to cases where the perturbing forces are small. Once applied only to comets and minor planets, special perturbation methods are now the basis of the most accurate machine-generated planetary ephemerides of the great astronomical almanacs. Special perturbations are also used for modeling an orbit with computers.

Cowell's formulation

Cowell's method. Forces from all perturbing bodies (black and gray) are summed to form the total force on body i (red), and this is numerically integrated starting from the initial position (the epoch of osculation).

Cowell's formulation (so named for Philip H. Cowell, who, with A.C.D. Cromellin, used a similar method to predict the return of Halley's comet) is perhaps the simplest of the special perturbation methods. In a system of ${\displaystyle n}$ mutually interacting bodies, this method mathematically solves for the Newtonian forces on body ${\displaystyle i}$ by summing the individual interactions from the other ${\displaystyle j}$ bodies:

${\displaystyle \mathbf {\ddot {r}} _{i}=\sum _{\underset {j\neq i}{j=1}}^{n}{Gm_{j}(\mathbf {r} _{j}-\mathbf {r} _{i}) \over r_{ij}^{3}}}$

where ${\displaystyle \mathbf {\ddot {r}} _{i}}$ is the acceleration vector of body ${\displaystyle i}$, ${\displaystyle G}$ is the gravitational constant, ${\displaystyle m_{j}}$ is the mass of body ${\displaystyle j}$, ${\displaystyle \mathbf {r} _{i}}$ and ${\displaystyle \mathbf {r} _{j}}$ are the position vectors of objects ${\displaystyle i}$ and ${\displaystyle j}$ respectively, and ${\displaystyle r_{ij}}$ is the distance from object ${\displaystyle i}$ to object ${\displaystyle j}$. All vectors being referred to the barycenter of the system. This equation is resolved into components in ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ and these are integrated numerically to form the new velocity and position vectors. This process is repeated as many times as necessary. The advantage of Cowell's method is ease of application and programming. A disadvantage is that when perturbations become large in magnitude (as when an object makes a close approach to another) the errors of the method also become large. However, for many problems in celestial mechanics, this is never the case. Another disadvantage is that in systems with a dominant central body, such as the Sun, it is necessary to carry many significant digits in the arithmetic because of the large difference in the forces of the central body and the perturbing bodies, although with modern computers this is not nearly the limitation it once was.

Encke's method

Encke's method. Greatly exaggerated here, the small difference δr (blue) between the osculating, unperturbed orbit (black) and the perturbed orbit (red), is numerically integrated starting from the initial position (the epoch of osculation).

Encke's method begins with the osculating orbit as a reference and integrates numerically to solve for the variation from the reference as a function of time. Its advantages are that perturbations are generally small in magnitude, so the integration can proceed in larger steps (with resulting lesser errors), and the method is much less affected by extreme perturbations. Its disadvantage is complexity; it cannot be used indefinitely without occasionally updating the osculating orbit and continuing from there, a process known as rectification. Encke's method is similar to the general perturbation method of variation of the elements, except the rectification is performed at discrete intervals rather than continuously.

Letting ${\displaystyle {\boldsymbol {\rho }}}$ be the radius vector of the osculating orbit, ${\displaystyle \mathbf {r} }$ the radius vector of the perturbed orbit, and ${\displaystyle \delta \mathbf {r} }$ the variation from the osculating orbit,

${\displaystyle \delta \mathbf {r} =\mathbf {r} -{\boldsymbol {\rho }}}$, and the equation of motion of ${\displaystyle \delta \mathbf {r} }$ is simply

(1)

${\displaystyle {\ddot {\delta \mathbf {r} }}=\mathbf {\ddot {r}} -{\boldsymbol {\ddot {\rho }}}}$.

(2)

${\displaystyle \mathbf {\ddot {r}} }$ and ${\displaystyle {\boldsymbol {\ddot {\rho }}}}$ are just the equations of motion of ${\displaystyle \mathbf {r} }$ and ${\displaystyle {\boldsymbol {\rho }},}$

${\displaystyle \mathbf {\ddot {r}} =\mathbf {a} _{\text{per}}-{\mu \over r^{3}}\mathbf {r} }$ for the perturbed orbit and

(3)

${\displaystyle {\boldsymbol {\ddot {\rho }}}=-{\mu \over \rho ^{3}}{\boldsymbol {\rho }}}$ for the unperturbed orbit,

(4)

where ${\displaystyle \mu =G(M+m)}$ is the gravitational parameter with ${\displaystyle M}$ and ${\displaystyle m}$ the masses of the central body and the perturbed body, ${\displaystyle \mathbf {a} _{\text{per}}}$ is the perturbing acceleration, and ${\displaystyle r}$ and ${\displaystyle \rho }$ are the magnitudes of ${\displaystyle \mathbf {r} }$ and ${\displaystyle {\boldsymbol {\rho }}}$.

Substituting from equations (3) and (4) into equation (2),

${\displaystyle {\ddot {\delta \mathbf {r} }}=\mathbf {a} _{\text{per}}+\mu \left({{\boldsymbol {\rho }} \over \rho ^{3}}-{\mathbf {r} \over r^{3}}\right),}$

(5)

which, in theory, could be integrated twice to find ${\displaystyle \delta \mathbf {r} }$. Since the osculating orbit is easily calculated by two-body methods, ${\displaystyle {\boldsymbol {\rho }}}$ and ${\displaystyle \delta \mathbf {r} }$ are accounted for and ${\displaystyle \mathbf {r} }$ can be solved. In practice, the quantity in the brackets, ${\displaystyle {{\boldsymbol {\rho }} \over \rho ^{3}}-{\mathbf {r} \over r^{3}}}$, is the difference of two nearly equal vectors, and further manipulation is necessary to avoid the need for extra significant digits. Encke's method was more widely used before the advent of modern computers, when much orbit computation was performed on mechanical calculating machines.

Periodic nature

Gravity Simulator plot of the changing orbital eccentricity of Mercury, Venus, Earth, and Mars over the next 50,000 years. The 0 point on this plot is the year 2007.

In the Solar System, many of the disturbances of one planet by another are periodic, consisting of small impulses each time a planet passes another in its orbit. This causes the bodies to follow motions that are periodic or quasi-periodic – such as the Moon in its strongly perturbed orbit, which is the subject of lunar theory. This periodic nature led to the discovery of Neptune in 1846 as a result of its perturbations of the orbit of Uranus.

On-going mutual perturbations of the planets cause long-term quasi-periodic variations in their orbital elements, most apparent when two planets' orbital periods are nearly in sync. For instance, five orbits of Jupiter (59.31 years) is nearly equal to two of Saturn (58.91 years). This causes large perturbations of both, with a period of 918 years, the time required for the small difference in their positions at conjunction to make one complete circle, first discovered by Laplace. Venus currently has the orbit with the least eccentricity, i.e. it is the closest to circular, of all the planetary orbits. In 25,000 years' time, Earth will have a more circular (less eccentric) orbit than Venus. It has been shown that long-term periodic disturbances within the Solar System can become chaotic over very long time scales; under some circumstances one or more planets can cross the orbit of another, leading to collisions.

The orbits of many of the minor bodies of the Solar System, such as comets, are often heavily perturbed, particularly by the gravitational fields of the gas giants. While many of these perturbations are periodic, others are not, and these in particular may represent aspects of chaotic motion. For example, in April 1996, Jupiter's gravitational influence caused the period of Comet Hale–Bopp's orbit to decrease from 4,206 to 2,380 years, a change that will not revert on any periodic basis.

 

Newton's laws of motion

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In classical mechanics, Newton's laws of motion are three laws that describe the relationship between the motion of an object and the forces acting on it. The first law states that an object either remains at rest or continues to move at a constant velocity, unless it is acted upon by an external force. The second law states that the rate of change of momentum of an object is directly proportional to the force applied, or, for an object with constant mass, that the net force on an object is equal to the mass of that object multiplied by the acceleration. The third law states that when one object exerts a force on a second object, that second object exerts a force that is equal in magnitude and opposite in direction on the first object.

The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), first published in 1687. Newton used them to explain and investigate the motion of many physical objects and systems, which laid the foundation for Newtonian mechanics.

Laws

Isaac Newton (1643–1727), the physicist who formulated the laws

Newton's first law

The first law states that an object at rest will stay at rest, and an object in motion will stay in motion unless acted on by a net external force. Mathematically, this is equivalent to saying that if the net force on an object is zero, then the velocity of the object is constant.

${\displaystyle \sum \mathbf {F} =0\;\Leftrightarrow \;{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=0}$

where:

• ${\displaystyle \mathbf {F} }$ is the net force being applied (${\textstyle \sum }$ is notation for summation),
• ${\displaystyle \mathbf {v} }$ is the velocity, and
• ${\textstyle {\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}}$ is the derivative of ${\displaystyle \mathbf {v} }$ with respect to time ${\displaystyle t}$ (also described as the acceleration.)

Newton's first law is often referred to as the principle of inertia.

Newton's first (and second) law is valid only in an inertial reference frame.

Newton's second law

The second law states that the rate of change of momentum of a body over time is directly proportional to the force applied, and occurs in the same direction as the applied force.

${\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}}$

where ${\displaystyle \mathbf {p} }$ is the momentum of the body.

Some textbooks use Newton's second law as a definition of force, but this has been disparaged in other textbooks.

Constant Mass

For objects and systems with constant mass, the second law can be re-stated in terms of an object's acceleration.

${\displaystyle \mathbf {F} ={\frac {\mathrm {d} (m\mathbf {v} )}{\mathrm {d} t}}=m\,{\frac {\,\mathrm {d} \mathbf {v} \,}{\mathrm {d} t}}=m\mathbf {a} ,}$

where F is the net force applied, m is the mass of the body, and a is the body's acceleration. Thus, the net force applied to a body produces a proportional acceleration.

Variable-mass systems

Variable-mass systems, like a rocket burning fuel and ejecting spent gases, are not closed and cannot be directly treated by making mass a function of time in the second law. The equation of motion for a body whose mass m varies with time by either ejecting or accreting mass is obtained by applying the second law to the entire, constant-mass system consisting of the body and its ejected or accreted mass; the result is^[10]

${\displaystyle \mathbf {F} +\mathbf {u} {\frac {\mathrm {d} m}{\mathrm {d} t}}=m{\mathrm {d} \mathbf {v} \over \mathrm {d} t}}$

where u is the exhaust velocity of the escaping or incoming mass relative to the body. From this equation one can derive the equation of motion for a varying mass system, for example, the Tsiolkovsky rocket equation.

Under some conventions, the quantity ${\displaystyle \mathbf {u} {\frac {\mathrm {d} m}{\mathrm {d} t}}}$ on the left-hand side, which represents the advection of momentum, is defined as a force (the force exerted on the body by the changing mass, such as rocket exhaust) and is included in the quantity F. Then, by substituting the definition of acceleration, the equation becomes F = ma.

Newton's third law

An illustration of Newton's third law in which two skaters push against each other. The first skater on the left exerts a normal force N₁₂ on the second skater directed towards the right, and the second skater exerts a normal force N₂₁ on the first skater directed towards the left.
The magnitudes of both forces are equal, but they have opposite directions, as dictated by Newton's third law.

The third law states that all forces between two objects exist in equal magnitude and opposite direction: if one object A exerts a force F_A on a second object B, then B simultaneously exerts a force F_B on A, and the two forces are equal in magnitude and opposite in direction: F_A = −F_B. The third law means that all forces are interactions between different bodies, or different regions within one body, and thus that there is no such thing as a force that is not accompanied by an equal and opposite force. In some situations, the magnitude and direction of the forces are determined entirely by one of the two bodies, say Body A; the force exerted by Body A on Body B is called the "action", and the force exerted by Body B on Body A is called the "reaction". This law is sometimes referred to as the action-reaction law, with F_A called the "action" and F_B the "reaction". In other situations the magnitude and directions of the forces are determined jointly by both bodies and it isn't necessary to identify one force as the "action" and the other as the "reaction". The action and the reaction are simultaneous, and it does not matter which is called the action and which is called reaction; both forces are part of a single interaction, and neither force exists without the other.

The two forces in Newton's third law are of the same type (e.g., if the road exerts a forward frictional force on an accelerating car's tires, then it is also a frictional force that Newton's third law predicts for the tires pushing backward on the road).

From a conceptual standpoint, Newton's third law is seen when a person walks: they push against the floor, and the floor pushes against the person. Similarly, the tires of a car push against the road while the road pushes back on the tires—the tires and road simultaneously push against each other. In swimming, a person interacts with the water, pushing the water backward, while the water simultaneously pushes the person forward—both the person and the water push against each other. The reaction forces account for the motion in these examples. These forces depend on friction; a person or car on ice, for example, may be unable to exert the action force to produce the needed reaction force.

Newton used the third law to derive the law of conservation of momentum; from a deeper perspective, however, conservation of momentum is the more fundamental idea (derived via Noether's theorem from Galilean invariance), and holds in cases where Newton's third law appears to fail, for instance when force fields as well as particles carry momentum, and in quantum mechanics.

History

Newton's First and Second laws, in Latin, from the original 1687 Principia Mathematica

The ancient Greek philosopher Aristotle had the view that all objects have a natural place in the universe: that heavy objects (such as rocks) wanted to be at rest on the Earth and that light objects like smoke wanted to be at rest in the sky and the stars wanted to remain in the heavens. He thought that a body was in its natural state when it was at rest, and for the body to move in a straight line at a constant speed an external agent was needed continually to propel it, otherwise it would stop moving. Galileo Galilei, however, realised that a force is necessary to change the velocity of a body, i.e., acceleration, but no force is needed to maintain its velocity. In other words, Galileo stated that, in the absence of a force, a moving object will continue moving. (The tendency of objects to resist changes in motion was what Johannes Kepler had called inertia.) This insight was refined by Newton, who made it into his first law, also known as the "law of inertia"—no force means no acceleration, and hence the body will maintain its velocity. As Newton's first law is a restatement of the law of inertia which Galileo had already described, Newton appropriately gave credit to Galileo.

Importance and range of validity

Newton's laws were verified by experiment and observation for over 200 years, and they are excellent approximations at the scales and speeds of everyday life. Newton's laws of motion, together with his law of universal gravitation and the mathematical techniques of calculus, provided for the first time a unified quantitative explanation for a wide range of physical phenomena. For example, in the third volume of the Principia, Newton showed that his laws of motion, combined with the law of universal gravitation, explained Kepler's laws of planetary motion.

Newton's laws are applied to bodies which are idealised as single point masses, in the sense that the size and shape of the body are neglected to focus on its motion more easily. This can be done when the line of action of the resultant of all the external forces acts through the center of mass of the body. In this way, even a planet can be idealised as a particle for analysis of its orbital motion around a star.

In their original form, Newton's laws of motion are not adequate to characterise the motion of rigid bodies and deformable bodies. Leonhard Euler in 1750 introduced a generalisation of Newton's laws of motion for rigid bodies called Euler's laws of motion, later applied as well for deformable bodies assumed as a continuum. If a body is represented as an assemblage of discrete particles, each governed by Newton's laws of motion, then Euler's laws can be derived from Newton's laws. Euler's laws can, however, be taken as axioms describing the laws of motion for extended bodies, independently of any particle structure.

Newton's laws hold only with respect to a certain set of frames of reference called Newtonian or inertial reference frames. Some authors interpret the first law as defining what an inertial reference frame is; from this point of view, the second law holds only when the observation is made from an inertial reference frame, and therefore the first law cannot be proved as a special case of the second. Other authors do treat the first law as a corollary of the second. The explicit concept of an inertial frame of reference was not developed until long after Newton's death.

These three laws hold to a good approximation for macroscopic objects under everyday conditions. However, Newton's laws (combined with universal gravitation and classical electrodynamics) are inappropriate for use in certain circumstances, most notably at very small scales, at very high speeds, or in very strong gravitational fields. Therefore, the laws cannot be used to explain phenomena such as conduction of electricity in a semiconductor, optical properties of substances, errors in non-relativistically corrected GPS systems and superconductivity. Explanation of these phenomena requires more sophisticated physical theories, including general relativity and quantum field theory.

In special relativity, the second law holds in the original form F = dp/dt, where F and p are four-vectors. Special relativity reduces to Newtonian mechanics when the speeds involved are much less than the speed of light.

Some also describe a fourth law that is assumed but was never stated by Newton, which states that forces add like vectors, that is, that forces obey the principle of superposition.