# The Physics, Astronomy and Mathematics of the Solar System

## Mathematics at the Frontier of Astronomy

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## Newton's Laws of Motion

Isaac Newton introduced a new approach to the study of the planets and their motions. The previous astronomers deduced their findings from observations. Newton on the other hand started with some general statements that related to the effects of forces on bodies and then showed how Kepler's laws applied. These laws not only describes the effect of the forces on Earth but also were applied to the heavenly bodies.

Sir Isaac Newton's three laws of motion enabled us to understand the motion of the bodies in the solar system. The laws relate the motion of the bodies to the force acting on them. The basic concepts of force and mass were defined.

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*

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The First law: The Law of Inertia.
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This law of inertia states that a body continues in its present state of motion unless it is acted on by a net external force. This law was opposite to the current belief but states that a body at rest is a special case. In this case the velocity is zero, with no force acting on it to cause it to move. A body will continue in a straight line until a force is applied to it. It was shown that this law was incomplete, because no consideration was given for effects of the force of the universe on that body. Since the body did not exist in a vacuum and the forces of the universe had an effect. It was concluded that to keep the body moving in a straight line for ever, the body must exist with nothing else. The law of inertia enables us to understand how the planets are able to maintain a constant path around the Sun, because of an unbalanced force is applied to it.

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Newton's Second Law: The Law of Forces
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The Law of forces relates to the change in the velocity of a body to an unbalanced force applied to the body. It states that the net force applied to a body is equal to the mass of the body times the acceleration caused in the body times the acceleration caused in the body by that force. The acceleration is in the same direction as the force. This law expressed as an equation is
*
F = ma
*
.

Acceleration measures the change in the velocity of that body. Velocity involves how fast an object is moving and also the direction of the motion. Acceleration is a result of a change in speed, in direction, or a change in both speed and direction. Quantitatively acceleration can be expressed as the amount of change in velocity divided by the interval of time.

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Let
*
Delta x = change in speed

Delta t = change in time

Then
*
a
*
= Delta x / Delta t

The law of force was also be used to measure the mass of an object. The mass of a body (the amount of matter) is defined as the resistance of the body to change in motion.

*
F =
*
*
ma
*
implies that two bodies have the same mass if the same force applied to the bodies causes the same acceleration.

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The Third Law: The Law of Reaction
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The law of reaction states that if one body exerts a force upon another body, the second body exerts a force equal and opposite to that force on the first body. This means that if a body exerts a force on another body causing it to move, the first body also moves. This movement is caused by the equal and opposite force exerted on it by the second body. Their acceleration can be calculated using Newton's Law of force. The smaller mass will experience a greater acceleration, while the larger mass will experience less acceleration. This is used to explain the Moon -Earth rotational system. The forces on both the Earth and the moon are the same strength, but the Moon's mass is less than that of the Earth, therefore the Moon does most of the moving.

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Newton's Law of Gravitation
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Newton explained the concept of gravity as the force that makes an object fall to the earth's surface, and used it to explain the planetary motions. He used the size of the Moon's orbit around the earth and its orbital period to calculate its acceleration.

Working with Kepler's law of planetary motion Newton found the force law required to explain the planetary motions. In this process he developed a new force law called the Law of Universal Gravitation and it is applicable throughout the universe. He also developed a new form of mathematics called calculus. He used this new mathematics to calculate the force of gravity between two bodies given as
^{
9
}
:

*
F
*
= G (m
_{
1
}
M
_{
2
}
/d
^{
2
}
), where

*
G
*
= the constant of gravitation

*
m 1
*
and
*
M2
*
= the masses of attracting bodies

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d
*
= the distance between the centers.

The force of gravity is always directed along the line joining the two bodies, therefore this law can be applied to a body on the Earth's surface as well as the Earth itself. The combination of the law of gravitation and the law of force is used to express the acceleration of a falling body at the Earth's surface in terms of the Earth's radius and mass. In the equation
*
F = ma
*
the gravitational force given by the law of universal gravitation as:

*
F
*
= G (mM e a r t h
_{
}
/ R e a r t h)

*
ma
*
= G (mM e a r t h
_{
}
/ R e a r t h2), and solving for a we get

*
a
*
= GM e a r t h
_{
}
/ R e a r t h2 =
*
g
*

The gravitation of earth is denoted by
*
g
*
. This has a value of
*
9.8 m/ sec
^{
2
}
*
. To test the gravitational law Newton used the data from the moon into the formula. It proved that the gravitational force decreases as the square distance between the bodies.

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Connections between Kepler's Laws of Planetary Motion and Newton's Laws of Motion and Gravitation
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Newton's laws of motion and gravitation can be to generalize Kepler's laws. Newton was able to predict a number of phenomena which were verified by observations.

By the law of reaction:

- 1. If two bodies are attracted to each other, both must be accelerated, therefore both are in motion.
- 2. Since the forces of gravity always lie along the line joining the two bodies, the common point about which they move must on that line.
- 3. The opposite forces of the bodies are equal therefore the acceleration of the bodies will be inversely related to their masses.
- 4. The larger body of mass will show the least motion, while the smaller body of mass will show the greater motion.
- 5. The common point around which the bodies move is called the center of mass. The product of the mass first body
(m) and its distance from the center of mass_{ 1 }(r) equals the product of the mass of the second body_{ 2 }(m) and its distance from the center of mass_{ 2 }(r). This is given as_{ 2 }m_{ 1 }r 1 = m_{ 2 r 2 }

When Newton's calculus is applied to his laws of motion and gravity Kepler's first law of planetary motion can be extended.

1. When two bodies under the force of mutual gravitation move on similar orbits about a common center of mass their orbits will always be of three kinds of curves: ellipses, parabolas, or hyperbolas. If a body has an elliptical orbit, then it repeats its path indefinitely. For both the parabolic orbits and the hyperbolic orbits since they represent open curves, the body in those orbits will have only one gravitational encounter with another body.

2. Kepler's law of equal areas represents a general law when Newton's theory is applied. The application reads; the line joining two bodies will sweep out equal areas in equal units of time. This law holds for any force law as long as there is a directional property, and, will apply whether the orbit is elliptical, parabolic or hyperbolic. The law implies that the bodies will move much faster in their orbits when the distance is smaller, and much slower when there are at a greater distance.

3. Kepler's harmonic law applies to elliptical or circular orbits and works only when one object orbits another. The relative orbit is made up of the sum of two similar individual orbits. Newton's introduce a new term the sum of masses and is given by the equation
*
(m
_{
1
}
+ m
_{
2
}
)
*

*P*where

^{ 2 }= a^{ 3 }
*
m
_{
1
}
*
and

*m*= the masses of the bodies expressed as solar masses

_{ 2 }
*
P
*
= the orbital period in years

*
a
*
= the semi- major axis of the relative orbits expressed in astronomical units. The sum of the masses expression is very useful in astronomy. It can be used to calculate the masses of stars.

Newton recognized that Kepler's laws were applicable for the case of two isolated interacting bodies, but failed in the case of the solar system where the attraction of the planets on each other causes deviations in their orbit. Kepler's laws would only work for all planets if they were the only an individual planet and the sun were present. The small forces exerted by the planets are minute when compared to the force of the sun. These small interplanetary forces are called perturbing forces, and the deviations caused on the orbits of the planets are called perturbations.