# Kepler’s law of planetary motion

### Kepler’s Law

Kepler gives three scientific laws about the motion of the planets around the Sun. Kepler’s law is the part of the Modern Astronomy and Physics.

Johannes Kepler published his firs two laws in 1609 and his third law in 1619.

Kepler’s three laws is given below;

**1. The planets orbit around the sun in ellipse at one of the two foci (foci is a plural word of focus. one focus, two foci).**

Mathematically a ellipse is represented by below formula;

Where is;

r= Distance from the su to the planet

p= Semi- latus rectum(semi-letus recto is the length which is half of the latus rectum).

ϵ= eccentricity of the ellipse

θ= angle of the planet’s current position from it’s closest approach, as seen from the sun.

r,θ= polar coordinates

Special conditions;

A. At θ=0 (In Perihelion)

B. At θ=90, 270

r=p

C. At θ=180 (In Aphelion)

The area of an ellipse is;

Where is;

a= Semi major axis

b= Semi minor axis

**2.** **A line which joining a planet and the sun sweeps out equal areas during equal interval of time.**

- There orbital radius and angular velocity of the planet in elliptical orbit are vary.
- When planet is closer to the sun in its orbit then it travel faster and when it is farthest to the sun in its orbit then it travel slower.
- The areal velocity will be constant.

**3.** **The square of the orbital period of a planet is directly proportional to the cube of the ****semi major axis of its orbit.**

Every Kepler’s orbit is a conic section.

### Complete Explanation

Actually Kepler’s law is a improvement in Copernicus’s law.

If the eccentricity of the planetary orbits is zero then Kepler’s law agrees with Copernicus’s law;

- Planets orbit around the sun in circular orbit.
- The sun is the center point of the circular orbit of the planet.
- The speed of the planet in is orbit is constant.

Both rules gives a good approximately of the planetary motion but Kepler’s law gives better observation.

Kepler’s correction in Copernicus’s law;

- Planets is not orbit around the sun in circular orbit but orbits in elliptical orbit.
- The sun is not the center point of the planetary orbit but it is a focal point of the planet’s elliptical orbits.
- The speed of the planet in its orbit is not constant. neither the linear speed nor the angular speed but area speed of planet in its orbit is constant.

Issac Newton improves the Kepler’s law which give us more accuracy.

Issac Newton computed acceleration of a planet according to the Kepler’s firs and second law;

- The direction of the acceleration is towards the sun.
- The magnitude of the acceleration is inversely proportional to the square of the planet’s distance from the sun.

Newton also tell about the force of the planet in its orbit;

- Every planet in our solar system is attracted toward the sun.
- The force on the planet is directly proportional to the mass of the planet and inversely proportional to the square of the planets distance from the sun.

So, Issac Newton assume;

- All bodies in our solar system are attract each other.
- The force is in between two bodies is directly proportional to the product of their mass and inversely proportional to the square of the distance of the planets from the sun.

## Eccentricity of the planets

**Eccentricity –** eccentricity is the ratio of the distance from the center to a focus divided by the length of the semi major axis. It represented by e.

**Perihelion** – perihelion is the closest point of a planet to the sun in it’s orbit.

**Aphelion** – aphelion is the farthest point of a planet to the sun in it’s orbit.

There are four conditions for eccentricity(e)-

- If e=0 then planet’s orbit will be circular orbit.
- If 0<e>1 then planet’s orbit will be elliptical orbit.
- And If e=1 then planet,s orbit will be parabolic escape orbit.
- If e>1 then planet’s orbit will be hyperbolic orbit.

**1. Mercury – **Mercury’s eccentricity is e=0.2056 which is greatest orbital eccentricity. The perihelion distance of the Mercury is 46 million km. and the aphelion distance is 70 million km. Mercury is the nearest planet to the sun. So it is the fastest planet in our solar system which complete its one round around the sun in 88 earth days.

**2. Venus – **Venus’s eccentricity is e=0.007 which is lest eccentricity in our solar system. It means it has a perfect circular orbit. The perihelion distance of the Venus is 107 million km and the aphelion distance is 109 million km. Venus is the second nearest planet to the sun. It take 224.7 days to orbit around the sun. Venus rotates so slowly in its orbit(a day on the Venus is equal to 1 year on the earth. It orbit clockwise.

**3. Earth – **Earth’s eccentricity is e=0.017. The perihelion distance of the Earth is 147 million km. and the aphelion distance is 152 million km. Earth is the third nearest planet to the sun. It takes roughly 365.256 days to orbit around the sun.

**4. Mars **– Mars’s eccentricity is e=0.093. The perihelion distance of the Mars is 207 million km and the aphelion distance is 249 million km. Mars is the forth nearest planet to the sun. It takes 687 days to complete its orbit.

**5. Jupiter – **Jupiter’s eccentricity is e=o.o48. The perihelion distance of the Jupiter is 741 million km and the aphelion distance is 778 million km. Jupiter is the fifth nearest planet to the sun. It takes 4330 earth days(11.86 earth years) to complete its orbit.

**6. Saturn **– Saturn’s eccentricity is e=0.056. The perihelion distance of the Saturn is 1.35 billion km and the aphelion distance is 1..51 billion km. Saturn is the sixth nearest planet to the sun. It takes 29.7 earth years to complete its orbit. This planet discovered in 1610. It means Saturn totally orbited 13 times since then. And the earth totally orbited 400 times since then.

**7. Uranus **– Uranus’s eccentricity is e=0.047. The perihelion of the Uranus is 2.75 billion km. and the aphelion is 3 billion km. Uranus is the 7th nearest planet to the sun. It takes 84.3 earth years to orbit around the sun.

**8. Neptune **– Neptune’s eccentricity is e=0.009. The perihelion distance of the Neptune is 4.45 billion km. and the aphelion distance is 4.55 billion km. Neptune is the 8th nearest planet to the sun. It takes 164.8 earth years to complete its orbit.

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