Kepler-90 e

Are Kepler-90 d, e, and f in resonance?

The orbital periods of Kepler-90 d, e, and f are approximately 59.74 days, 91.94 days, and 124.91 days, respectively. If the innermost planet d's period is normalized to 2, the ratios of the three periods become roughly 2:3.078:4.182, which is close to 2:3:4. However, the ratio of periods alone cannot confirm that the three planets are in resonance. A 2024 study showed that the angles representing the positions of the three planets did not stay within a fixed range, and there was no evidence that the three planets were connected in a single resonance.

• Approximately 91.94 days Orbital period of e measured through repeated transits
• 2 : 3.078 : 4.182 Actual ratio of d, e, and f after normalizing d's period to 2
• About 2.66 Earth radii Radius of e calculated from transit depth and the star's size
• About 10 times Earth's level Starlight energy calculated to reach e

How far are their orbital periods from a 2:3:4 ratio?

If the periods are exactly in a 2:3:4 ratio, then while d completes six orbits, e completes four, and f completes three. In reality, the period ratio of d to e is about 1.539, slightly more than the exact 1.5, and the period ratio of e to f is about 1.359, slightly more than the exact 1.333. Because of this small difference, the times when the three planets meet in the same positional relationship shift slightly.

Calculating the time it takes for this positional change to repeat using only the current orbital periods gives about 23 years. This is much longer than the approximately 4 years during which Kepler observed this star. Therefore, even if the orbital periods of the three planets are precisely known, it is difficult to directly confirm from observational data alone whether the positional relationships repeat within a certain range or continue to change.

Orbital periods close to a 2:3:4 ratio provide clues that resonance might be possible. To confirm resonance, it is also necessary to know whether the positional relationships of the three planets repeat within a certain range.

How can we confirm whether the three planets are in resonance?

Astronomers combine the positions of d·e·f in a fixed ratio to represent them as a single angle, which is called the Laplace angle. If the three planets are in a resonance state connected by gravity, this angle does not pass through all directions but oscillates back and forth within a certain range. Conversely, if it continues to vary from 0 degrees to 360 degrees, it is not considered a complete Laplace resonance even if the orbital periods are close.

In a 2014 discovery study, the authors suggested the possibility of a three-planet resonance by looking at the period relationships of d·e·f. Subsequent researchers calculated how the Laplace angle changes over time by inputting various masses, orbital shapes, and initial positions. In the 2024 study, this angle did not remain within a fixed range but continued to vary, and the 2:1 resonance between d and f was not confirmed. Therefore, currently, it can only be said that the periods of the three planets are close to 2:3:4.

1. Period measurement From repeated transits of the three planets, 59.74 days, 91.94 days, and 124.91 days were obtained.
2. Integer ratio comparison It examines how close 2:3.078:4.182 is to 2:3:4.
3. Position relationship calculation Input the mass, orbital shape, and initial position to track the change in the Laplace angle.
4. Judgment Resonance is determined based on whether the angle stays within a certain range or changes in all directions.

What can similar period ratios tell us about how the planets formed?

Planets can move through the gas and dust around a star and gather near simple period ratios. If the gas and dust disappear or the planets exchange energy with each other, period ratios slightly away from resonance may remain. It is also possible that they were accidentally formed in similar ratios from the beginning. The current periods alone cannot tell which process actually occurred.

e is located between d and f, so it is influenced by both planets' gravity. To calculate this influence accurately, the actual masses and orbital shapes of the three planets must be known. The masses of d, e, and f listed currently are estimates based on statistics from other planets of similar size and are not results from direct measurement of their gravity.

Can Kepler-90 e’s orbital period reveal its environment?

The radius of e is about 2.66 times that of Earth. The mass of about 7.56 Earth masses listed is not a directly measured value but an estimate based on statistics of other planets of similar size. With just these two numbers, the amounts of rocky core, easily evaporating substances like water, and hydrogen-helium atmosphere cannot be known. Even planets of the same size can have different actual masses if their internal materials differ.

e orbits its star at about 0.42 AU, approximately 62.8 million km away, receiving about 10 times the energy Earth gets from the Sun. Assuming a simple reflection of starlight and heat transfer, its temperature is about 450 K, roughly 177 degrees Celsius. This value is not the actual temperature measured in the atmosphere or on the surface. To know the seasons, one would need the tilt of the rotation axis, the shape of the orbit, and the atmosphere's heat transfer, none of which have yet been measured for e.

The planet shown on the page is not an observational photo but an illustration for understanding. The period and radius of e are based on transit observations, but the mass and temperature of approximately 177 degrees Celsius come from statistics of other planets and simple heat calculations. The complete three-planet resonance of d, e, and f has not been confirmed.

What more must we observe to determine whether the planets are in resonance?

To determine more precisely whether the three planets are in resonance, astronomers need new transits observed many years after Kepler and must compare their actual and predicted times. They can also measure how the planets' mutual gravity shifts each transit, or how their gravity makes the star move, to estimate the planets' actual masses. Those masses and orbital data can then be used to recalculate the Laplace angle and test whether e links d and f in a resonance.

Sources

• NASA Exoplanet Archive — Kepler-90 e Parameters and Mass Provenance
• Astronomy & Astrophysics — The Planetary System to KIC 11442793
• Monthly Notices of the Royal Astronomical Society — Analysing the Dynamics of the Kepler-90 Planetary System
• The Astronomical Journal — Dynamics of Tightly Packed Planetary Systems
• The Astrophysical Journal — Probabilistic Mass–Radius Relation
• NASA Science — Kepler Transit Method
• NASA Exoplanet Archive — Kepler Data Products Overview