In the 6th chapter of The Assayer, published in 1623, Galileo notes:
Philosophy [i.e. natural philosophy] is written in this grand book — I mean the Universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.
With the revolutionary spirit that characterizes his thinking, Galileo explains in a few words that the laws that govern the functioning of the Universe are mathematical. If we do not understand that physical phenomena speak to us in the language of mathematics, they will remain incomprehensible. The researcher will seem to be uselessly wandering around in a dark labyrinth with no way out.
The successes that contemporary science obtains thanks to the power of mathematics would probably amaze Galileo. Not for the method, given that he had correctly understood that the “book” of the cosmos is written in numbers and geometric shapes, but for the richness and complexity of the information that you can now extract from a few observational data. The characterization of planetary systems hundreds of light-years away from us is an example of this power when applied to light from stars.
One case that best illustrates the amount of information that can be extracted from the photometric and spectroscopic analysis of stellar light concerns the system formed by the star KELT-9 and its planet KELT-9b, a hot, indeed very hot, Jupiter.
KELT-9 is the hottest main-sequence star among those known to be orbited by planets . Cataloged with the spectral type B9.5–A0, it has an effective temperature of 10,170 ± 450 K and is 53 times brighter than the Sun. The mass and radius are 2.53 and 2.36 times the respective solar values. To be clear, it is a star similar to Vega and Altair, but even larger and brighter. And much more distant: located in the constellation Swan, the distance obtained from the parallax angle measured by the Gaia satellite is 670 light-years .
A study published in Nature in 2017 described the discovery through the transit method of KELT-9b, a gas giant in a tight orbit around this bright and hot star. Thanks to a vast amount of photometric and spectroscopic data collected over the years, the research authors managed to obtain a series of valuable information. KELT-9b is, first and foremost, a huge and massive planet. The radius equals 1,891 Jovian radii, which is 135,190 km (about 21 times Earth’s radius). The mass corresponds to 2.88 Jovian masses, the equivalent of 915 Earth masses.
This giant orbits at a frantic speed of 254.3 km/s, making a complete revolution around the star in less than a day and a half (1.481 Earth days, to be precise). It also travels an unusual polar orbit, very different from the orbits of the solar system’s planets, which are roughly coplanar to the equatorial plane of the Sun. With an orbit’s semi-major axis of just 0.034 astronomical units (5.18 million km), KELT-9b is hit by an ocean of stellar radiation. The incident flux is calculated in 61 million watts per square meter (6.11 × 10⁷ W/m²). It is a radiation flux over 44,700 times greater than that which hits the Earth at the top of the atmosphere!
The planet’s equilibrium temperature, inferred from insolation, is 4,050 ± 180 K. But even higher is KELT-9b’s day hemisphere temperature, calculated at around 4,600 K . This means that this fiery giant is hotter than many stars, hotter than, say, all red dwarfs. You have to get to the spectral type K4 to find stars with the same temperature as KELT-9b.
The fierce bombardment of ultraviolet radiation to which the planet is subjected has dramatic effects on its atmosphere. Especially in the day hemisphere, temperatures are so high that molecules dissociate. In fact, studies based on transmission spectroscopy have revealed that KELT-9b’s atmosphere is rich in atomic species, including iron, titanium, and magnesium. The entire planetary atmosphere is subject to continuous erosion by stellar radiation. The calculations indicate that KELT-9b’s atmospheric mass loss is in the order of millions of tons per second!
The already considerable amount of information available on KELT-9b and its star has recently been further enriched, thanks to a new study published in June 2020 in The Astronomical Journal, based on the accurate photometric data collected by NASA’s TESS space telescope. Between July and September 2019, TESS observed 27 transits of KELT-9b. The salient feature of the light curves acquired by the telescope during those transits is that they are asymmetric. The dip in the star luminosity is slightly more marked during the initial phase of each transit than during the final phase. How do you explain this asymmetry?
First of all, it must be considered that KELT-9, the star, is a fast rotator. The minimum rotational velocity is 111.4 ± 1.3 km/s, much higher than that of the Sun, which is just 2 km/s. The enormous centrifugal force generated by the fast rotation produces two effects: 1) it changes the shape of the star in an oblate spheroid, wider at the equator than at the poles; 2) surface gravity is less at the equator than at the poles, where more mass is compressed in a given space (a phenomenon known as gravity darkening).
As a consequence of this deformation and the gravity gradient associated with it, KELT-9 is hotter and brighter at the poles than at the equator. And not by a little: the difference is about 800 K. Here is what happens during the transits of the planet, according to the explanation reported in the study published in June:
When KELT-9 b transits its host star, it blocks a certain amount of light depending on whether it is transiting near the bright poles or dim equator. In the case of a misaligned orbit, the planet blocks varying intensities of light throughout its transit, resulting in an asymmetric transit. KELT-9 b’s transit ingress is deeper than its egress, meaning the planet begins its transit near KELT-9’s hot pole and moves toward its cooler equator.
If the orbital plane of KELT-9b had been aligned with the star’s equatorial plane, no asymmetry in the transit would have been noticed. Instead, the manifest asymmetry of the transits recorded by TESS betrays the polar orbit followed by the planet. Not only that: it allowed the study authors to obtain precise information on the star’s rotational velocity and the inclination of its axis, the flattening generated by the centrifugal force, the temperature gradient between star’s equator and the poles, and the planet’s precise orbital inclination relative to the star rotation axis. It is an incredible amount of information, considering that we see neither the star disk nor that of the planet from Earth. The only observable elements are the variations in time of KELT-9’s brightness and its radial velocity, measured through Doppler spectroscopy: a magnificent demonstration of the explanatory power of mathematics applied to physical phenomena, under the Galilean idea that the laws of Nature are written in the language of numbers and geometric shapes.
Here are the primary data obtained from the analysis of the asymmetric transits observed by TESS. The inclination of the star’s rotation axis relative to an observer on Earth is 52°, with a margin of uncertainty of +8 and −7 degrees. It means that the rotation axis is seen from the Earth inclined diagonally (the inclination would be 90° if the axis were vertical relative to us and 0° if it were horizontal). KELT-9’s actual rotation period is 16 hours, with an uncertainty of +5 and −4 hours. The star’s equatorial radius is 2.39 solar radii, that is, 1.633 million km. The oblateness, indicated with the Greek letter ζ, is equal to 0.089 ± 0.017, from which it is deduced that KELT-9’s polar radius is 148,000 km shorter than the equatorial radius. The Sun, in comparison, is an almost perfect sphere.
But the most interesting data are those concerning the effects of the variable insolation that the planet gets during its orbit. According to the study authors, when KELT-9b approaches the poles of the star, it gets a radiation flux 10% higher in the ultraviolet and 1–2% in visible light. Due to these flux variations, temperatures in the upper atmosphere of the planet are subject to radical changes during each orbit. The differences in temperature, in turn, probably produce impetuous winds, which blow at speeds of thousands of kilometers per hour: a phenomenon quite common in hot Jupiters such as KELT-9b. But the most singular effect is that the short year of this giant planet sees two summers and two winters happening in a matter of hours: a consequence of the star’s gravity darkening. In this regard, we read the explanation provided by the study authors:
The effect of gravity darkening on a planet’s insolation can be compared to the insolation of a planet with an eccentric orbit. In both scenarios the planets receive varying amounts of flux throughout their year, which can drastically impact climate. However, the frequency of changing flux is twice per orbit for gravity darkening versus once per orbit in eccentricity. […] Gravity darkening likely plays a more significant role than eccentricity for the insolation of planets such as KELT-9 b because hot Jupiter orbits are typically nearly circular.
Since during an orbit, KELT-9b passes in succession near both poles of the star, which are much hotter and brighter than the equator, its year sees two summers alternating with two winters. “Winters,” so to speak, of course, given that, in any case, KELT-9b’s temperatures are above 4,000 K.
In conclusion, it would be exciting to see KELT-9b up close and study the effects on its climate of the deadly radiation flux to which it is continually exposed. Unfortunately, the enormous interstellar distances prevent us from satisfying such scientific curiosities. But the knowledge about exoplanets accumulated over the past twenty-five years allows us to form at least a general picture of the varieties of existing planetary systems. A fiery giant like KELT-9b, trapped in a polar orbit a short distance from an oblate and very hot star, shows us an example of a star/planet system utterly different from the Solar System. Such a configuration induce one to wonder what the evolutionary history of this planet was. What gravitational interactions brought the planet to that polar orbit? An issue about which, for now, we can only make educated guesses.
 There are 4,197 exoplanets currently confirmed. The very few main-sequence stars hottest than KELT-9 reported in the NASA database are orbited by brown dwarfs, i.e., bodies with at least 13–14 Jovian masses, which are considered failed stars rather than planets.
 The equivalent of 6.34 million billion km (6.339 × 10¹⁵ km, to be precise).
 Since KELT-9b’s orbit is very tight, the planet is most likely tidally locked, always showing the same hemisphere to the star as the Moon does with the Earth.