An imperfect theory
It was convenient to assume that the planets orbited around the sun in perfect circles when early astronometers first began modeling the solar system. It was an idea that goes back to Plato and remained commonplace up until the 17th Century.
The heliocentric model assumes that the Earth revolves around the sun in a perfect circle. Yet there is a problem with this model when you observe the motion of planets closely. Here is a sequence of images of the sun (taken from the earth) over the course of a year. Pay close attention to the size:
Notice the size of the sun is gradually changing? This is not a result of the sun growing and shrinking. This is phenomena is a result of the changing distance between the earth and sun.
Johannes Kepler (1571 – 1630) was a german astronomer who realized that circular orbits wouldn’t work while investigating the orbital motion of Mars in close detail. Kepler writes about his discovery to a fellow astronomer (David Fabricius) on October 11th 1605:
“So, Fabricius, I already have this: that the most true path of the planet [Mars] is an ellipse, which Dürer also calls an oval, or certainly so close to an ellipse that the difference is insensible.”
An ellipse can have different values for its width and height. This means the radius will change depending on the angle through the full orbit. One simple way to think about an ellipse is the addition of two different sized circles which defines an x and y coordinate respectively. In the example below the x-coordinate comes from the larger circle and the y-coordinate comes from the smaller circle. Convince yourself of this:
It's very important to notice that a circle is a set of points which are a fixed distance from the center. However an ellipse is a set of points a certain distance from two points (called the foci). These are two points on the major axis such that the sum of the distance between any point on the ellipse and both foci is constant.
Below is an interactive illustration. You can click and drag the foci to change the shape. Notice that the green and blue lines always add up to the same distance:
Which leads to Kepler’s first law:
The orbit of every planet is an ellipse with the sun at one of the two foci.
Preparing to animate
To draw an elliptical orbit, we define the x-axis radius (a) and the y-axis radius (b). The major axis is the larger of the two, the minor axis, the smaller. Notice that if a=b, then the equations are the same as the ones we used for a perfect circle.
x = a x cos(θ)
y = b x sin(θ)
We can now define an ellipse with three properties: its centre, its major axis and its minor axis.
Next let's review the equations of an ellipse.
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- If the sun is at on of the foci, where is the other foci of the earth? Why is it there anyway?(4 votes)
- Kepler's laws state that the sun is located at one of the two foci. The other focus is known as an 'empty' focus. Both the foci are located on the major or long axis of the ellipse.(3 votes)
- how many laws did kepler have(2 votes)
- Kepler had 3 laws.
#1-Planets move in ellipses with the sun at one focus.
#2-A planet moves faster when closer to the sun and slower when farther away.
#3-The time a planet takes to orbit the sun is mathematically related to its average distance from the sun.
Hope this helps.(10 votes)
- I have a question regarding the third law. I understand the third law mathematically. and V sq=GM/r , this is where i dont understand, if v is larger, wont the object move out of its trail instead of closer? but in the equation when V increase r decrease.(4 votes)
- If a planet has to revolve in fixed orbits, the gravitational force between the sun and the star should balance the centrifugal force of the earth which is constant motion. So,
G*m1m2/r^2=m2*v^2/r [m1 is the mass of the sun and m2 is the mass of earth]
Seeing that G*m1 is a constant, by ratio and proportion,
v^2 is proportional to 1/r or v^2 is inversely proportional to r.
I was a little surprised at my deductions. Then, the speed of the earth revolving around the sun at perihelion should be more compared to the earth revolving at aphelion. Are the calculations correct? Then k should be approximately and numerically equal to 1.327*10^20.(2 votes)
- Nice Question... It had me thinking for some time and when I worked out the math this is what I came up with...
The formula for the velocity of an object at some distance r from the Sun is:
v = sqrt[GM*(2/r - 1/a)]
Where G is the universal gravitational constant, M is the mass of the Sun, and a is the planet's semimajor axis.
At perihelion, Earth's distance from the Sun is r=a(1-e) and at aphelion, it's r=a(1+e).
G=6.673*10-11 N m2/kg2
So plugging in the numbers, the speed at perihelion is 30,300 m/s and at aphelion it's 29,300 m/s. So, YES the speed of the earth revolving around the sun at perihelion is greater compared to the earth revolving at aphelion.
My calculations are based on Ellipse Co-ordinate geometry.
I hope this helped!
Feel free to ask any more questions...(1 vote)
- how do we explain the second law of kepler to others in simple words(1 vote)
- here is a diagram. A1 and A2 are both areas equal in size, not shape, because the planet moves slower when it is further from the sun.
- How the Planet can have 2 foci ?(3 votes)
- I believe it has something to do with the fact that as the planets get closer to the Sun, they speed up due to the Sun's mass---they end up being "flung" at a higher velocity and that is why their orbits are eclipses and not circles (hence, two foci, as ellipses have)(1 vote)
- according to keplers second law why do the planets when they are closer to the sun?(1 vote)
- Kepler's second law provides that orbiting bodies sweep out the same area for equal amounts of time. Thus, for closer parts of their orbit, they have to sweep out more distance of their orbit to match the area for a section swept out when its further away.
- I've learnt that Kepler's third law states that T squared = a cubed. Can someone tell me where I can get a derivation for this, and also a derivation of Newton's Law of Universal Gravitation?(1 vote)