02 Kepler’s Third Law

02 Kepler’s Third Law#

Aim#

To show empirically that Kepler’s third law is true.

Subjects#

  • 1L20 (Orbits)

  • 8A10 (Solar System Mechanics)

Diagram#

../../../../../_images/figure_056.png

Fig. 135 .#

Equipment#

  • Graph on overhead sheet, \(T=f(a)\), \(T\) and a both scaled logarithmically.

  • Table with data of the planetary system (see Sources).

Presentation#

The graph is projected by means of an overhead sheet. The relationship with the table of planetary data is elucidated. Clearly can be observed that the data fit on a straight line in such a double logarithmic graph. The slope of this line (\(p/q\)) equals 1.5. This is the relationship of the powers in Kepler’s third law: \(T^2\propto a^3\)

Explanation#

Kepler’s third law states \(T^2=c \times a^3\) with \(c\) a constant. Taking logarithms on both sides, we can also write:

\[2\log T = \log c + 3\log a\]

and:

\[\log T = \frac{1}{2}\log c + \frac{3}{2}\log a\]

So when T and a are graphed logarithmically (with \(x\)– and \(y\)-decades equally spaced), we see a line whose slope (\(\frac{3}{2}\)) is the power-relationship in the original function.

Simulations#

ISSUE: SIMULATION NEEDED

Sources#

  • Mansfield, M and O’Sullivan, C., Understanding physics, edition 1998, pag. 106-107 and 741 (planetary data).

  • BINAS tabellenboek, vijfde druk, tabel 31.

  • McComb, W.D., Dynamics and Relativity, edition 1999, pag. 72-74.

  • Roest, R., Inleiding Mechanica, vijfde druk, pag. 257-258.

  • Stewart, J, Calculus, edition 1999, pag. 867.

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