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# Log-Log plot and Kepler's third law.
Suppose you have collected some data points $(x_{i},y_{i})$, and you make a plot of them. Sometimes the data have a linear relationship between $x$ and $y$. But sometimes it could be a power relation say $y=Ax^p$ for some power $p$. Then how might we determine $p$?
Instead of looking at the graph of $(x_{i},y_{i})$, look at the plot of $(\log x_{i},\log y_{i})$. If there is power relationship $y=Ax^p$, then taking logarithm of both sides give $$
\log y=\log A+p\log x
$$
![[---images/---assets/---icons/question-icon.svg]] If points $(x,y)$ satisfy some power law $y=Ax^p$, what would the plot of $(\log x,\log y)$ look like? What is the slope? And what is the vertical intercept of this log-log plot?
![[---images/---assets/---icons/question-icon.svg]] Kepler in his later years discovered his **third law** of planetary motion, on the relation between the length of one year for each planet and their average distance to the sun. Kepler obtained this law by playing around with the numbers of the data he has!
Here are some data available to Kepler in 1618:
|Planet|Mean distance(in AU)|One-year(in days)|
|---|---|---|
|Mercury|0.389|87.77|
|Venus|0.724|224.70|
|Earth|1|365.25|
|Mars|1.524|686.95|
|Jupiter|5.20|4332.62|
|Saturn|9.510|10759.2|
Try to discover a function $T(a)$ between a planetary year $T$ (in days) and its mean distance $a$ (in AU) if there is some power relation between $T$ and $a$.
Here are some modern values (2018)
|Planet|Mean-distance|Year|
|---|---|---|
|Mercury|0.38710|87.9693|
|Venus|0.72333|224.7008|
|Earth|1|365.2564|
|Mars|1.52366|686.9796|
|Jupiter|5.20336|4332.8201|
|Saturn|9.53707|10775.599|
|Uranus|19.1913|30687.153|
|Neptune|30.0690|60190.03|