=== My Story with "Amateur Astronomer" ===
Last year, when I subscribed to the 2012 issues of Amateur Astronomer, I only ordered a half-year subscription because I knew I would be starting university soon. Consequently, I haven’t read any new issues of Amateur Astronomer since June of this year. Having gone through the two months of summer break, plus September and October, it has been nearly four months without it. I thought I had already grown accustomed to life without the magazine.
About a week ago, I re-purchased the issues from the past four months on the magazine’s Taobao store. On the afternoon of the 18th, I was reunited with it. That night, I was suddenly moved, overwhelmed by a flood of emotions. Although it had been so long, the feeling of reading it again was so familiar and warm. I used to think that astronomy was just a side interest of mine, like biochemistry, but in that moment I realized that I truly love astronomy, and neither time nor distance can diminish that love! At that moment, I decided that I must pursue an astronomy-related profession—even though I am currently just a mathematics student!
========== Lower Limit of Planetary Orbital Period ==========
(2012.10.25: zwhzjh pointed out an error in the perturbation force formula. I have corrected the calculation formula. Previously, a factor of 2 was missing. Additionally, in the final empirical verification, I previously prioritized the "reasonableness" of the result over scientific methodology; this has now been corrected. Further comments are welcome.)
The topic of this article is something I discovered by chance while reading Amateur Astronomer. Before the discovery of exoplanets, it was generally believed that gas giants like Jupiter should have orbital periods of ten years or more. Therefore, when Swiss astronomers Michel Mayor and Didier Queloz discovered the first exoplanet, they could hardly believe their own findings, because this Jupiter-like planet had an orbital period of only 4.2 days! However, after confirmation, it was indeed an exoplanet, overturning past assumptions. I pursued this research with great interest, attempting to derive the lower limit of the orbital period for a planet of a given density.
Readers might want to estimate first: what physical quantities will it be related to? The planet’s mass? The host star’s mass? Or something else?
The truth is—it depends only on the density of the planet itself!
Of course, the results here are derived simply by comparing perturbation forces and gravitational forces, and can only serve as an order-of-magnitude estimate. A precise estimation would naturally require considering various other factors.
First, assume the planet has mass m, radius r, and density \rho; the host star has mass M, the distance between them is R, and the orbital period is T.
For a planet to exist, the gravitational force of its own mass on objects at its surface must be greater than the perturbation force exerted by the host star; otherwise, the host star would tear it apart. We make the following assumption: for stable existence, the planet’s self-gravity must be at least two orders of magnitude (10^2) greater than the host star’s perturbation force. This ratio is somewhat subjective, but it possesses a certain degree of rationality. After all, if it were at the critical point, any random fluctuations could compromise its stability, so it must be far from the threshold.
Thus, we can easily list the following:
10^2 \frac{2GMr}{R^3} < \frac{Gm}{r^2} \quad \text{(Our assumption)}
m = \frac{4}{3}\pi r^3 \rho \quad \text{(Mass and density)}
\frac{GM}{4\pi^2} = \frac{R^3}{T^2} \quad \text{(Kepler's Third Law)}
Combining these, it is not difficult to solve for the lower limit formula of the orbital period:
T > \sqrt{\frac{600\pi}{G\rho}} \approx \frac{20 \text{ days}}{\sqrt{\rho}}
In the final formula, the density must be in \text{kg/m}^3 to obtain the correct result.
========= Verification with Exoplanets =========
The final result surprisingly cancels out all other quantities, leaving only the density and some constants. Furthermore, this tells us that planets with very short periods generally have very high densities. White dwarf binaries can even orbit each other in just a few minutes, which alone allows us to conclude that the density of white dwarfs is extremely high. We can verify that all currently known planets satisfy this result. Jupiter and Saturn in our solar system go without saying; their periods are over ten years, so they certainly comply. According to online data, the fastest exoplanet discovered so far is SWEEPS-10. Its orbit is only 740,000 miles (about 1.19 million kilometers) from its host star, its radius is approximately 1.24 \pm 0.23 times that of Jupiter, and its "year" is only 10 hours. Scientists estimate it has at least 1.6 times the mass of Jupiter. We can also estimate this using the formula above. 10 hours is approximately 0.42 days, which yields \rho \approx 2286 \text{ kg/m}^3. Since Jupiter’s density is \rho \approx 1326 \text{ kg/m}^3, and using the minimum radius data, the mass ratio of SWEEPS-10 to Jupiter is approximately:
\frac{2286}{1326} \times 1.01^3 \approx 1.78
This lower limit is consistent with scientists’ estimates. Although the result is slightly on the higher side, it reflects the reasonableness of our estimation to a certain extent! That is to say, while the initial order-of-magnitude assumption was subjective, it was reasonable. Of course, there is a bit of "post-hoc fitting" behind the scenes here ^_^
I wonder what other thoughts you might have?
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