By Joshua N. Winn, Princeton University
On a regular chart, the horizontal axis is mass, in units of Earth masses, and the vertical axis is radius, in units of Earth radii. If we remake this chart with logarithmic axes, then the horizontal axis still tells us the mass, but now each tick mark represents a factor of 10. Astrophysicists use logarithmic charts to help understand things that range over many orders of magnitude.
Logarithmic Set of Axes
Let’s understand what dataset looks like on the logarithmic set of axes.
There are 4 different groups, differing in the relationship between mass and radius. For the lowest-mass objects, we can draw a straight line through the points up until about twice the mass of the Earth, at which point the pattern changes; the size starts going up more rapidly with mass. The line connecting the dots has a steeper slope. Then at around a 100 Earth masses, this relationship changes again; it flattens out. And finally, at around 20,000 Earth masses, the size starts rising again with mass, more rapidly than in any other part of the chart.
To get oriented, consider the data points for all the planets in our solar system. Together, they span the first 3 groups, and the Sun is in the fourth group. In each of these 4 zones, we can fit the data, at least approximately, with a straight line.
Regular Vs Logarithmic Chart
On a regular x-y chart, a straight line means that y = ax + b, where a is the slope of the line, and b is a constant, the y-intercept. That’s a linear relationship.
But on a logarithmic chart, a straight line means that there’s a linear relationship between the logs of the variables: log x = a log y + b; or in this case, the log of the radius equals a times the log of the mass, plus a constant. And what does that tell us? It’ll be clearer if we solve for R. We can do that by taking the inverse log of both sides, that is, we’ll take 10 to the power of each side. On the left, we have 10 to the log R which is just R, as we wanted. On the right, we have 10 a log M + b.
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The First Zone
When we have a sum in the exponent, we can split it into a product. We can rewrite the right side as 10a log M x 10b. And another fact is that a log M is equal to log of M to the power a. So, the anti-log of a-log-M is Ma. The net result is that R is proportional to M to the a power.
That kind of relationship is called a power law. One variable is proportional to another one raised to some power.
If we measure the slope, the value of a, in each of these 4 zones, we note that for the lowest mass objects, the slope is about 1/3. That means R is proportional to M to the 1/3 power: the cube root of M.
This low-mass regime is closest to the one where we have direct experience: small things, like rocks and boulders. So, what do we expect for the relation between radius and mass of everyday objects? Well, it should depend on the density of whatever material it’s made of.
Let’s use the Greek letter rho for density. Rocks have a certain characteristic density, around 2 or 3 grams per cubic centimeter. And mass is equal to density times volume. For a spherical object, the volume is 4/3 pi times the cube of the radius. So, we expect M to be proportional to R cubed, or, inverting that, R is proportional to M to the 1/3 power. Which is what we see in the chart!
The Second Zone
In the second zone, we find a slope of about a 1/2; R is proportional to M to the 1/2 power, which is bigger than 1/3. That means that the more massive objects have larger radii than we would have expected, if they all had the same density. So, it must be that the more massive objects are less dense.
And this is true. These are gaseous planets, like Uranus, Neptune, and Saturn. In this zone, the planets have a lot of hydrogen and helium gas, very lightweight elements, in addition to rock.
The Third and Fourth Zones
In the third zone, the line is horizontal: the slope is zero. The size hardly changes at all, even when we increase the mass by a factor of 100. In everyday life, when we pack more mass onto a ball, the ball gets bigger. But apparently, this is not the case for balls between 100 and 10,000 Earth masses.
Our own Jupiter is in this zone, as are lots of exoplanets. What must be happening here is that the more massive objects are much denser than the less massive versions. Part of the reason that they’re increasingly dense is gravitational compression. These objects are so massive that their own gravity compresses them to higher densities than usual.
There’s no single name for all the objects in this zone. Sometimes we call them ‘Jovian planets’, although toward the higher mass end, the traditional term is ‘brown dwarfs’.
Finally, we have the highest mass objects, for which we get a slope of about one, which means that for these guys, radius is proportional to mass.
These are stars—objects for which the gravitational compression is so strong that nuclear fusion ignites at the center, creating lots of heat and pressure. This same nuclear fusion also produces the light that stars are famous for; it’s what makes stars shine.
Common Questions about Logarithmic Charts
Gaseous planets have a lot of hydrogen and helium gas, very lightweight elements, in addition to rock.
The more massive objects are much denser than the less massive versions. Part of the reason that they’re increasingly dense is gravitational compression. These objects are so massive that their own gravity compresses them to higher densities than usual. Thus, their size hardly changes at all, even when we increase the mass by a factor of 100.
Stars are objects for which the gravitational compression is so strong that nuclear fusion ignites at the center, creating lots of heat and pressure. This same nuclear fusion also produces the light that stars are famous for; it’s what makes stars shine.