Which Planets on Which Orbits?
Thus, by rearranging, it is possible to arrive at a “required” mass figure for a planet occupying a given orbit:
This equation requires that an orbital period (in Earth years) be specified for the planet in order to calculate a mass for the planet; thus, the mass of the planet is not solely dependent on the orbital distance, but upon both orbital parameters: how far from the star it orbits, and also the time it takes to do so. Assuming the orbital distance remains the same, any change in the chosen orbital period will cause a corresponding, usually drastic inverse change in the calculated necessary mass of the planet (see below).
Let’s look at the case of Jupiter: it orbits the Sun at 5.202 AU and its mass is 0.0009546 that of the Sun. We know empirically that its orbital period is 11.8618 Earth years, but let’s calculate it using the long form of the equation:
So, what happens if we change Jupiter’s orbital period by a minuscule amount to, say, 11.0 Earth years exactly?
What if we halve its orbital period to 5.926 Earth years (leaving it in its current orbit)?
As can be seen, changes in the orbital period have effects on the resultant calculated mass for the planet, and the larger the change in the orbital period, the more drastic the change in the planet’s mass, if the orbital distance remains the same. In this case, the planet has actually grown to three times the size of the star it's orbiting!!!
Note, also, that setting the orbital period to any value greater than Jupiter’s known orbital period results in an impossible negative mass for the planet; for instance, let’s raise the orbital period minutely, to exactly 12.0 Earth years:
This is true for all known planets: a planet’s “default” orbital period is the upper limit for its semi-major axis. In order for the planet to orbit more slowly, the semi-major axis of the orbit must increase. We can calculate the size the orbit would have to have by rearranging our equation to:
This shows that the relationship between orbital distance, orbital period, and the mass of the orbiting body is a delicate one:
• Reducing Jupiter’s orbital period by a mere 0.852 years increased its mass by over 170 times,
• Reducing its orbital period by half caused Jupiter to become three times more massive than the Sun; and,
• Increasing Jupiter’s orbital period without increasing its orbital axis caused it to have an impossible negative mass.
Does this mean that Jupiter cannot orbit at, say, Venus’s distance from the Sun? No, of course it can: in fact, many exoplanets orbit far closer to their stars. As mentioned in the previous blog, Kepler 42c, for instance, has a semi-major axis of ~0.00585 AU from its star, or less than one-sixtieth the distance of Mercury from the Sun.
What it does mean, though, is that to assume that Jupiter would have the exact same orbital period as Venus on the exact same orbit as Venus may not be a valid assumption.
Using the simple axis-period relation:
Because mass isn’t involved in the short form of the equation, a Jupiter-mass planet (318 times more massive than Venus), would be assigned the same orbital period if assigned to the same orbit.
Instead, let’s calculate Jupiter’s orbital period on Venus’ orbit using the long form of the equation:
As I have noted elsewhere, while it is technically possible for any planet to occupy any orbit, exoplanet systems so far discovered seem to show a tendency for inner-system planets to be terrestrials and outer-system planets to be giants. (See the illustrations at Cosmic Diary and Centauri Dreams.)
Small star and big planet
Using the simple version of the equation first:
Using the long version:
Note: ANY planetary mass ≥ 0.0002282 times the mass of its parent star results in an orbital period shortfall of 0.01142% when using the short form of the equation instead of the long form of the equation. This amounts to a difference of one hour per year. For the Solar System, this is ~23.9% of the mass of Jupiter, 79.85% of the mass of Saturn, and about 4.43 times the mass of Neptune.
A Note About The 1.0 Earth-Year Orbital Period
Under ANY other conditions the simple formula will not return a correct orbital period for a planet orbiting its star(s) at 1.0 AU.
Thus, any time the value of MS not 1.0, the full equation must be used, or at least the equation in the form:
Note also that the greater the magnitude of the value of MS from 1.0—either smaller or larger—the greater will be the error in the calculated orbital periods; in other words, ignoring an MS of 1.1 will produce a smaller error than would ignoring an MS of 2.5.
Assume a central star—Orban—with a mass of 1.5 times that of the Sun.
Orbiting Orban at 1.0 AU is a planet—Saýas—with the same mass as the Earth.
If we use the simple equation to calculate Saýas' orbital period, we will arrive at a value of 1.0 Earth-years, because:
The mass of the star (MS) and the mass of the planet (MP) need to be expressed in terms of the Sun. The value of MS in the equation above is straightforward: it is 1.5, because we've already said that Orban is 1.5 solar units in mass. However, Saýas' mass also needs to be expressed in terms of that of the Sun.
Fortunately, we've said that Saýas' mass is the same as Earth's so we need only divide the mass of the Earth in grams (5.972E+27) by the mass of the Sun in grams (1.989E+33) to get Saýas' (and Earth's) mass in solar units:
Thus, we'll use 3.0025E-06 as MP in the above equation, which will look like this:
If Saýas' mass were, perhaps, 0.62 Earth-masses, then we'd need to calculate its mass in terms of Earth's mass expressed in solar units:
What to do?
Well, if we don't care that Saýas' orbital period is shorter than Earth's, we can stop here.
However, if we want to be sure that Saýas' orbital period is 1.0 Earth-years, then we need to calculate its semi-major axis based on that fact (and the sum of the masses of Orban and Saýas).
Our derived equation, from above, for calculating a semi-major axis based on the orbital period and the masses of the bodies, is:
- The central star(s) of our system have a mass (or combined mass) which is anything other than 1.0 solar-masses; and,
- We know we want a planet (regardless of its mass) to have an orbital period of 1.0 Earth-years,
- Use the full version of the equation to calculate what are the orbital periods for the planets orbiting at the calculated semi-major axes; or,
- Use the derivation of the full version to calculate at what semi-major axis a planet with an orbital period of 1.0 Earth-years must orbit. This result, then, is what we would shoe-horn into whatever other orbits we obtained from one of the formulas.
A Further Note about 1.0 solar luminosity
HOWEVER . . .
It should be noted that an orbital period of 1.0 Earth-years around a star (or stars) with any single or combined mass other than 1.0 solar-masses does not guarantee that the stellar irradiance on that planet will be the same as that which the Earth receives from the Sun at 1.0 AU.
This is because a star of any mass other than 1.0 solar-masses will also have a luminosity other than 1.0 solar-luminosity.
Recall that both gravity and electromagnetic radiation obey the inverse-square law:
For example, if we use mass in place of force in the above equation:
We can also use the same form of the calculation to determine intensities of stars at give distances.
A star of 1.0 solar-luminosity at a distance of 1.0 AU produces the same insolation as the Sun does on the Earth:
A star of 1.0 solar-luminosity (L = 1.0) at 0.5 AU (d = 0.5), produces four times the insolation:
Because changes in the mass or luminosity of your central star(s) affect their luminosity or mass, respectively, the only way to have your planet possess an orbital period of 1.0 Earth-years and receive the same insolation from its star(s) as the Earth does from the Sun is for your planet to have a mass of 1.0 Earth-masses and your star(s) to have a single or combined mass of 1.0 solar-masses, and a single or combined luminosity of 1.0 solar-luminosities.
Here's what I mean: Let's say you want to have a star (Dahel) of 0.85 solar-masses, but you want your planet (Aldat) to have a mass of 1.0 Earth-mass and an orbital period of 1.0 Earth-years. At what distance must Aldat orbit Dahel? 
However, it will not have a stellar irradiance (insolation) equal to that received by the Earth from the Sun, because a star of 0.85 solar-masses has a radius and temperature of:
If we change the intensity of Dahel to ensure 1.0 solar-insolation at 0.94727 AU distance, then the mass of the star changes, and the orbital period of Aldat is no longer 1.0 Earth-years.
Here are the calculations:
The necessary luminosity for 1.0 solar-insolation at 0.94727 AU is:
If the desire is to have a planet with an orbital period of 1.0 Earth-years and a mass of 1.0 Earth-masses, then it will be necessary for it to orbit a star or stars with a single or combined luminosity of 1.0 solar-luminosities and a single or combined mass of 1.0 solar-masses.
However, if any of the following are true:
- You have a planet with a mass totaling a significant percentage of the mass of your star(s);
- You have a central star (or stars) with a total mass of anything other than 1.0 solar units;
- You have a planet you know must have an orbital period of precisely 1.0 Earth-years;