Документ взят из кэша поисковой машины. Адрес оригинального документа : http://star.arm.ac.uk/~meb/files_2011aug24/kepler.doc Дата изменения: Fri Aug 26 02:55:23 2011 Дата индексирования: Tue Oct 2 05:49:45 2012 Кодировка: Поисковые слова: п п п п п п р п р п р п р п р п р п р п р п р п р п р п р п

Kepler's Laws

Kepler was an assistant to Tycho Brahe. Brahe had made extensive
observations of the known planets and had accurately mapped the motion of
these planets as they passed across the sky for a number of years.

For more information go to the below link
http://csep10.phys.utk.edu/astr161/lect/history/kepler.html

Kepler deduced three laws, based on these observations.

The orbits of the planets are ellipses with the sun at one of the foci.

Look at the Planets in the Orrery. Mercury is the obvious planet, but Enke
and Halley show this in a much more exaggerated way. Look at the distance
between the discs. The discs are set at equal time intervals (one step
being 16 days). For Encke compare the distance between the discs and
complete the table below. Remember disc 0 for Encke is buried underneath
disc 5 for Mercury. This is an approximation as the path is not straight.
Try to include the curvature of the orbit in your distance measurement.

Name of planet____________________________


|Disc numbers (e.g. 1 |Distance (m) |Speed |
|to 2) | |(distance (m)/step |
| | |increment) |
| | | |
| | | |
| | | |
| | | |
| | | |

Name of planet____________________________

|Disc numbers (e.g. 1 |Distance (m) |Speed |
|to 2) | |(distance (m)/step |
| | |increment) |
| | | |
| | | |
| | | |
| | | |
| | | |

What do you notice about the speed of the celestial body as it gets closer
and further from the sun?
Why?
Do the other planets follow the same pattern? Look carefully at Jupiter and
Saturn and see if your ideas are sound.



Kepler's Laws

All bodies sweep out equal areas in equal times

If we look at the planets we have already seen that the planets change
speed in their orbit. Further from the sun they slow down, closer they
speed up. Kepler proposed that the area swept out by the planet in any
given time period should be the same. The two sectors in the below right
diagram have the same area. i.e.
Area SAB = area SCD

Because the orbit is an ellipse the area is not that easy to calculate.
However we can approximate the area by saying it is part of a circle, to do
this we need to take discs that are adjacent to each other.
Method 1
Using a tape measure or ruler, measure the distance of the planet from the
sun tile at two points. Take an average of this distance (not strictly
correct procedure). Measure the distance between the two discs (remember it
is a curve) and the radius of the orbit.

For a circle of circumference 2pr and area pr2 then the area of our sector
of arc length s, is approximately:




|Disc numbers (e.g. |Distance apart (s) |Orbital radius (r) |Area (s*r/2) |
|1 to 2) |(m) |(m) | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |


A second method uses the angles written on the discs. We can use the angle
"f" as written each disc. Using a simple ratio the area of the arc is given
by




|Disc numbers (e.g. |Change in angle ?f |Orbital radius (r) |Area (?f * ?r2 |
|1 to 2) | |(m) |/360) |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |

In essence these are the same method. Do the results agree?
Do other planets give data that agree with Kepler's law?
Kepler's Laws

Kepler's 3rd law looks at the period of the orbit (T) and the radius of the
orbit (r)

The period squared is proportional to the cube of the average distance from
the sun (T2 ? r3)

Using the data on the discs find the period for each planet and the average
radius for each planet's orbit.

Now plot a graph of T2 vs r3 and if Kepler is correct we should obtain a
straight line through the origin.

|Planet |Period of |Average |T2 |r3 |
| |orbit |distance |(Years2) |(AU3) |
| |(T)(years) |from the sun| | |
| | |(r) | | |
| | |(AU) | | |
|Mercury | | | | |
|Venus | | | | |
|Earth | | | | |
|Mars | | | | |
|Ceres | | | | |
|Jupiter | | | | |
|Saturn | | | | |






















-----------------------
Image courtesy of
www.kepler.nasa.gov

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]