Exoplanet Stuff

[lpetrich]

I improved the above tally to include discovery methods. For some reason, transit timing variations are not included among the observation-method flags in the data file.

Transit - tran	2983
Radial Velocity - rv	886
Transit - rv,tran	750
Microlensing - micro	124
Imaging - ima	50
Transit - rv,tran,obm	20
Radial Velocity - rv,tran	18
Eclipse Timing Variations - etv	16
Transit Timing Variations - tran	11
Transit Timing Variations -	8
Imaging - ast,ima	7
Pulsar Timing - pul	7
Orbital Brightness Modulation - obm	5
Orbital Brightness Modulation - rv,obm	3
Transit Timing Variations - rv	3
Pulsation Timing Variations - ptv	2
Radial Velocity - rv,ast	2
Radial Velocity - rv,ima	2
Radial Velocity - rv,obm	2
Radial Velocity - rv,tran,obm	2
Astrometry - ast	1
Disk Kinematics - dkin	1
Imaging - rv,ima	1
Orbital Brightness Modulation - rv,tran,obm	1
Transit - rv	1
Transit - tran,etv	1
Transit - tran,obm	1

 

[lpetrich]

I decided to sort the detections out by discovery and followup.

Method	Discovery	Followup	Total
Transit	3756	32	3788
Radial Velocity	912	779	1691
Microlensing	124	0	124
Imaging	58	2	60
Orbital Brightness Modulation	9	25	34
Transit Timing Variations	22	0	22
Eclipse Timing Variations	16	1	17
Astrometry	1	9	10
Pulsar Timing	7	0	7
Pulsation Timing Variations	2	0	2
Disk Kinematics	1	0	1

 

[lpetrich]

Some methods are only used for discovery, like microlensing, transit timing variations, pulsar timing, pulsation timing variations, and disk kinematics (from gaps in protoplanetary rings).

Some are mostly used for discovery, like transits, imaging, and eclipse timing variations, and others mostly for followup, like orbital brightness modulation and astrometry. Radial velocity is about half-half.

 

[lpetrich]

[2109.10422] Precise Masses and Orbits for Nine Radial Velocity Exoplanets
IoW_20220131 - Gaia - Cosmos

A feature of Gaia's measurements that may look odd is how its measurements are shown as lines instead of as points. This is from how Gaia works. It measures angles between celestial bodies by rotating between them, using a full rotation as a reference. This makes positions very precise along one direction but not nearly as good perpendicular to it. That is what makes those lines.

That aside, astrometry complements radial velocity by being able to measure an orbit's inclination, something also true of transits.


All these methods are persistent with one exception,  Gravitational microlensing That is from a star or a planet going in front of a distant star and lensing the star's light. The article shows a typical event's light curve -- a single peak with a half-maximum width of 50 days and and a maximum amplification of 3.

If the source is directly behind the lens, then its light will form a ring around the lens, an  Einstein ring Its angular radius is

\( \displaystyle{ \theta_E = \sqrt{ \frac{4GM}{c^2} \frac{D_s - D_l}{D_s D_l} } } \)

M = lens mass, D_s and D_l are the source and lens distances.

The amplification A is

\( \displaystyle{ A(u) = \frac{u^2 + 2}{u \sqrt{u^2 + 4}} ,\ u = \frac{\theta}{\theta_E} } \)

where θ is the angle between the source and the lens. For constant linear motion as a function of time t, the angle is

\( \displaystyle{ \theta = \sqrt{ (\theta_0)^2 + (\theta' \cdot (t - t_0))^2 } } \)

with angle minimum θ₀ at time t₀ and long-distance rate θ'.

Since a lensing event is a one-time event for each planet, one has to observe a large number of events to get good statistics.

 

[lpetrich]

All the other methods are observations of persistent effects, effects that can be repeatedly observed over time.

Of these methods, disk kinematics is the opposite of gravitational microlensing: observation of something that is stationary in time: gaps in protoplanetary disks caused by planets.  Protoplanetary disk -  PDS 70

THe others are persistent and time-varying.

• Transient state: Microlensing
• Stationary state: Disk Kinematics
• Persistent and time-varying: Transit, Radial Velocity, Imaging, Orbital Brightness Modulation, Transit Timing Variations, Eclipse Timing Variations, Astrometry, Pulsar Timing, Pulsation Timing Variations

Imaging is direct detection: taking a picture of the planet.

Transits are a sort of opposite of imaging. Transits are mini-eclipses, planets blocking their stars' light.

Orbital brightness modulation is three effects:

• Reflection and emission modulation -- planets going through illumination phases as they orbit their stars
•  Relativistic beaming -- focusing of light in the direction of motion
• Ellipsoidal variations -- planets making tides on their stars, tides that give those stars an American-football-shaped distortion

Detection methods by method:

• Direct: imaging, illumination phases
• Inverse: transits
• Gravitational:
  • On light: microlensing
  • On orbiting dust: disk kinematics
  • By other planets: transit timing variations
  • On the planet's star:
    • Line-of-sight offset: eclipse timing variations, pulsar / normal-star-pulsation timing variations
    • Line-of-sight velocity: radial velocity, relativistic beaming
    • Perpendicular offset: astrometry
    • Tidal distortion: ellipsoidal variations

Eclipse timing variations are timing variations of the eclipses of eclipsing binary stars.

 

[lpetrich]

I should digress as to how we measure the masses of planets' stars, something necessary in most methods for finding the planets' masses. Most stars with known planets are loners, meaning that they don't have another star nearby that can help in measuring its mass. But most of them are main-sequence stars, meaning that one can get approximate masses from their luminosities and spectral types. One uses the  Mass–luminosity relation - luminosity is approximately some power of the mass: about 2.3 for red dwarfs, 4 for Sunlike stars, and 3.5 for blue giants.

But main-sequence stars gradually brighten over the time in that state, because of the accumulation of helium in their cores, making greater opacity, and thus more brightness for the light to get out of it.

The faint young Sun problem - Feulner - 2012 - Reviews of Geophysics - Wiley Online Library notes an approximate equation for the Sun:

\( \displaystyle{ \frac{L}{L_0} = \frac{1}{1 + \frac25 \left( 1 - \frac{t}{t_0} \right) } } \)

where t is the time since the Sun's origin, t₀ is the time since the Sun formed, 4.57 billion years, L is the Sun's luminosity, and L₀ is the Sun's present-day luminosity, 3.85*10²⁶ watts.

The Sun will exit the main sequence about 5 billion years from now, meaning that its luminosity will range from 5/7 to 5/3 of its present-day value, 0.72 to 1.67, over a factor of 2. That causes a potential inaccuracy of measurement of the Sun's mass from its luminosity, an inaccuracy of around 10%.

It turns out that precision photometry can help us, the same precision photometry that has made possible the discovery of hundreds of exoplanets. This photometry enables the observation of starquakes, much like sunquakes, and these quakes' frequencies provide clues as to the stars' interiors. In particular, how much helium the stars have accumulated, thus making possible age estimates, and with their luminosities their masses.

 

[lpetrich]

I now turn to what one can get of planets' orbits from these methods.

Planets observed with direct imaging are usually relatively distant, with long periods. That means that in the time that we have been able to observe them, they have not traveled over much of their orbits, meaning that we cannot have a good fix on their orbits.

The best-known case of such planets is the four known planets of star  HR 8799 spectral type F0, mass 1.4 * Sun, luminosity 5 * Sun, age about 30 million years. That is deduced from the surface temperatures of those planets, about 1000 K, enough to glow in visible light. Those planets are all Jovian ones, and they orbit at outer Solar System to Kuiper Belt distances by Solar System standards.

Planets e, d, c, b (outward in reverse alphabetical order) have distances 16, 27, 41, and 72 AU, and periods 45, 100, 190, and 460 years. Their orbits are approximately circular and coplanar.

Planets b, c, and d were discovered in 2008, and previous observations of them were done in 1998 ("precovery"). Planet e was discovered in 2009. That is a time to present of 13 years for e and 24 years for b, c, and d. Assuming additional observations this year, the observed fractions of the planets' orbits are thus 0.29, 0.24, 0.13, 0.05.

Velocity estimates relative to e: 1, 1.36, 1.11, 0.80
Acceleration estimates relative to e: 1, 1.13, 0.49, 0.14
Acceleration-rate estimates relative to e: 1, 0.94, 0.21, 0.03

Acceleration rate may be an odd one to include, but there's a reason for it. An orbit is determined by six parameters: the position and velocity at some time, all in 3D. In practice, one finds "orbit elements" which are much more convenient for calculation and informative about the orbit. They are size, eccentricity, three orientation angles, and the position in the orbit at some time.

An observed position is projected, and thus has only two values, clearly not enough. One can hand-wave the distance from the star using statistics: (average actual) = (observed) * (pi/2)

An observed position and velocity are both projected, and thus have four values, also not enough.

Acceleration has the problem of what makes it: gravity.

avec = - GM*rvec/r³

Since what we observe is projected, we find GM/r³. If we know the mass of the star, we get the distance, thus giving five observed quantities. That is clearly not enough, and that is why we need the acceleration rate. It is

- GM*vvec/r³ + 3*GM*rvec*(rvec.vvec)/r⁵

Using GM/r³ from the acceleration, we find (rvec.vvec)/r². That gives us the remaining quantity, and thus all six orbit elements.

If one has the acceleration of the acceleration, one can do even better. One finds

- 2*(GM)²*rvec/r⁶ + 6*GM*vvec*(rvec.vvec)/r⁵ + 3*GM*rvec*v²/r⁵ - 15*GM*rvec*(rvec.vvec)²/r⁷

giving us v²/r², an additional measured value. One can then treat the star's mass as an additional orbit element and estimate it using that value and the previously-found ones.

 

[lpetrich]

In practice, one needs much of an orbit to get the acceleration of the acceleration, and one tries to fit the orbit directly. In the process, one may use the orbit period as an additional orbit element, equivalent to using the star's mass that way, but more convenient to calculate with. Measuring over several orbits often makes it the best-measured orbit element.

An interesting curiosity is a seeming violation of Kepler's First Law, that a planet's orbit around its star has the star at one of the foci of its bit. But in a tilted orbit, the star is away from the orbit foci. This is rather obvious for a circular orbit -- the star is at the center of the observed ellipse, not at its foci.

So one can use the orbit ellipse and the star's location relative to it to find most of the orbit elemets.


Everything that I've said about direct-imaging orbit determination also applies to astrometric orbit determination from precision position measurements on stars. There,

(star-displacement orbit size) = (total planet orbit size) * (planet mass) / ( (star mass) + (planet mass) )

So one needs to treat the orbit period as an additional orbit element.


All of this work is very familiar territory for astronomers, as they have observed binary stars for over two centuries. Sixth Catalog of Orbits of Visual Binary Stars — Naval Oceanography Portal lists over 2500 of them - the catalog itself: Sixth Orbit Catalog


 Orbital elements goes into detail. The three orientation ones are the inclination of the orbit to the coordinate equator, the longitude of the ascending node, where the orbit goes coordinate upward through the coordinate equator, and the argument of the periapsis, the angle from the ascending node to the periapsis position in the direction of the orbit.

For a binary star or an exoplanet, the coordinate equator is the plane of the sky.

 

[lpetrich]

Having considered transverse offsets (direct imaging of planet, astrometry on star), I consider line-of-sight offsets of the star: pulsar timing, normal-star pulsation timing, eclipse timing variations, radial velocity, relativistic beaming.

They involve a projected orbit size (m/M)*a*sin(i)

where m = planet mass, M = total mass, a = semimajor axis, and i = inclination.

With an estimate of the star's mass, we can find the planet's projected mass, m*sin(i).

This can disambiguate orbits from direct detection or star astrometry.

Eclipse timing is for close eclipsing binaries that are orbited by a planet. This planet not only makes an offset of the stars' positions, it also pulls the stars alternately ahead and behind, making an additional effect. But that orbit-perturbation effect is much smaller than the lever-arm effect, and I estimate a relative size of (astr/apln)⁴ where apln is the size of the planet's orbit around the stars and astr is the size of the stars' orbit.

Relativistic beaming is from light being focused in the direction of the star's motion. For velocity v << c, it's a weak effect. Combined with Doppler shifting, it causes a total relative luminosity change of 4*(vproj/c) where vproj is the projected velocity along the line of sight. The 4 has these contributions: 2 from aberration of the star's light in each transverse dimension, 1 from Doppler shifting, and 1 from photon-emission rate change. For redshift z, it is (1+z)^-4.

Tidal distortion makes ellipsoidal variations, and that happens in two ways: changing the cross-sectional area of the star and changing which average surface temperature that one observes. Due to lower surface gravity, the football distortion's poles will be slightly cooler than the distortion's equator, and thus less bright. The effect has a size of some numerical factor times

(m/M)*(R/r)³*sin(i)²*sin(p)²

where m and M are the planet's and star's masses, r and R are the planet's distance and star's radius, i is its orbit inclination, and p is its position angle relative to the plane-of-sky ascending node. So we get a measurement of

m*sin(i)²

 

[lpetrich]

One can be more systematic about the series expansion of the Kepler-Newton two-body solution:

• Lagrange/Gibbs F and G Solutions
• f- and g-Series -- from Eric Weisstein's World of Physics
• High order F and G power series for orbit determination.

The f and g are in

\( {\mathbf r(t)} = f(t) {\mathbf r_0} + g(t) {\mathbf v_0} \\ {\mathbf v(t)} = {\dot f}(t) {\mathbf r_0} + {\dot g}(t) {\mathbf v_0} \)

where subscript 0 means at reference time (epoch) t = 0: f(0) = 1 and g(0) = 0. The dot indicates a time derivative, much like '. We get these differential equations for f and g:

\( \dot f = \dot f_{int} - g s \\ \dot g = \dot g_{int} + f \)

where the {int} denotes an "internal" derivative, a derivative of the contents of the expression itself. It turns out that each step of f and g can be expressed in polynomials of the following three variables:

\( \displaystyle{ s = \frac{\mu}{r^3} ,\ p = \frac{ {\mathbf r} \cdot {\mathbf v} }{r^2} ,\ w = \frac{ v^2 }{ r^2 } } \)

where μ = G*M = mass in gravitational units, the  Standard gravitational parameter. These Lagrange variables have derivatives that are polynomials in them with integer coefficients:

\( \displaystyle{ {\dot s} = - 3 s p ,\ {\dot p} = w - s - 2p^2 ,\ {\dot w} = - 2p (s + w) } \)

 

[lpetrich]

Here is a calculation up to derivative number 5:

Deriv number	f	g
0	1	0
1	0	1
2	-s	0
3	-3*p*s	-s
4	-s*(2*s + 15*p^2 - 3*w)	6*p*s
5	15*p*s*(2*s + 7*p^2 - 3*w)	-s*(8*s + 45*p^2 - 9*w)

 

[lpetrich]

It's easy to read off of that calculation to see what one can find for a partial orbit. By derivative number:

• 0: radius vector
• 1: velocity vector
• 2: s -- M/r^3
• 3: p -- (r.v)/r^2
• 4: w -- v^2/r^2

For a circular orbit, p = 0, w = s.

Which derivative:

• Direct observation of a planet (transverse):
  • 4 -- orbit, star mass
  • 2 -- assumed circular orbit, star mass
• Astrometry of a star, assumed star mass (transverse):
  • 5 -- orbit, planet mass
  • 4 -- assumed circular orbit, planet mass
• Pulsation, radial velocity of a star, assumed star mass (line-of-sight):
  • 6 -- orbit, projected planet mass
  • 4 -- assumed circular orbit, projected planet mass

For direct observation, all one needs for a circular orbit with a star-mass observation is observation of the planet's motion. For all the rest, one need more of a planet's orbit.

A rough estimate can be found by solving for the orbit fraction f in fⁿ = e for the nth derivative having error e. That gives us f = e^1/n.

Exoplanet observations are not very precise, so I will consider relative error sizes 0.1 and 0.01. For error 0.1, one gets fractions 0.1, 0.32, 0.46, 0.56, 0.63, 0.68, and for error 0.01, one gets fractions 0.01, 0.1, 0.22, 0.32, 0.40, 0.46. So for most of the observation methods, one will need to observe over much of the planet's orbit before being able to determine that orbit with any confidence.

As to why the star observations need higher derivatives than the planet ones, it's because one can refer the planet positions to the planet's star, while one has no such reference for the star itself -- one has to subtract out its position and velocity in interstellar space.

 

[lpetrich]

I now come to the remaining method: transit timing variations.

These are caused by another planet pulling the transiting planet alternately forward and backward in its orbit.

The mathematics is very complicated, however, though astronomers have done that math since the 18th cy., and one can make some simple estimates of the sizes of the effects.

One can do numerical integration, but that's not very informative, and it's mainly useful when one wants to improve rough-estimate orbits. It was also not very practical before the development of electronic computers.

1951AJ.....56...38E

Eckert, W. J. Numerical theory of the five outer planets.

The equatorial rectangular coordinates of the planets Jupiter, Saturn, Uranus, Neptune and Pluto from 1653 to 2060 were determined by numerical integration. All the mutual attractions of the five planets and the sun were included and the integration, at forty-day intervals, was made with an accuracy of fourteen decimals.

The constants of integration were determined by comparison with all available observations. Two preliminary integrations were made for this purpose, the second covering the interval from 1780 to 1940.

Those calculations were done on the  IBM SSEC - Selective Sequence Electronic Calculator - in service from 1948 to 1952. It was the last large electromechanical computer ever built. It used 12,500 vacuum tubes for its arithmetic-logical unit and 8 registers, and about 21,400 electromechanical relays for its main memory, for both programs and data.

 History of computing hardware -  History of computing hardware (1960s–present) -  History of timekeeping devices -  Timeline of time measurement inventions

The first known computing mechanism is likely the Antikythera Machine of a little over 2,000 years ago. It used a clocklike gear mechanism for calculating the positions of the planets. Geared clocks in general became common in the late Middle Ages. That's very specialized computing hardware, however, and more general hardware became common in the late 19th cy.: adding machines. Generalized electromechanical computers were developed around World War II, with all-electronic ones and stored-program ones following soon after. Transistors followed in 1955, and integrated circuits in the early 1960's. The first commercial CPU on a chip was the Intel 4004 in 1971, and single-chip CPU's became more and more capable, with faster operation, larger blocks of data, and more on-chip memory.

Returning to exoplanets, I note Refining the Transit-timing and Photometric Analysis of TRAPPIST-1: Masses, Radii, Densities, Dynamics, and Ephemerides - IOPscience -- using a lot of numerical integrations to find the masses and orbits of the system's planets.

 

[Loren Pechtel]

Scratch 4 exoplanets: https://www.syfy.com/syfy-wire/bad-astronomy-massive-exoplanets-really-stars

(Registration wall.)

 

[lpetrich]

One can get more insight by doing analytic calculations, even though that requires very complicated mathematics. In fact, this analytic approach was done before large-scale numerical calculations became easily feasible.

 Perturbation (astronomy)

• Analytic calculations: general perturbations
• Numerical integration: special perturbations

In the analytic approach, can do the calculations in two ways: directly finding the offset from the unperturbed orbit, and finding the variations in the orbit elements. The second one is more difficult to set up, but it can be more informative about the behavior of the perturbed orbit.

In fact, one can do a hybrid approach, finding differential equations for the orbit elements then averaging them over individual orbits. That gives some equations for long-term orbit-element behavior that one then solves.

For nearly circular orbits, one can rearrange the elements as follows:

• h = (ecc) * sin(pericenter longitude)
• k = (ecc) * cos(percenter longitude)
• p = sin((inc)/2) * sin(ascending-node longitude)
• q = sin((inc)/2) * cos(ascending-node longitude)

To lowest order, the differential equations for long-term perturbations are linear in the h, k, p, and q values, and the equations' solution is an eigensystem problem, like the solution of a multicomponent oscillation problem. One gets a quasiperiodic solution, a set of oscillations with various periods, with the planets' elements sharing those periods but having different coefficients for them.

If one plots the solutions for each planet's h, k, p, and q, one finds a complicated loop-the-loop behavior, though in some cases, some of the oscillations are dominant.

One can next calculate the effects of these oscillations on planets' spin-axis orientations, and one finds the Milankovitch astronomical cycles, cycles in obliquity (spin-orbit inclination), precession (relative to orbit pericenter), and eccentricity. Note that the eccentricity cycle modulates the strength of the precession cycle. For the Earth, these cycles fit well the comings and goings of the continental glaciers over the last 2.5 million years.

 Milankovitch cycles - the Earth's orbit eccentricity is now 0.0167, but its average value is 0.025, and it is sometimes as large as 0.05. Likewise, the Earth's orbit's orientation wobbles about 1 degree or so from its average orientation. The latter effect causes an obliquity oscillation of about 1 degree, and we are currently at 23.5d inclination, close to the average value.

The Earth's spin precession has a period of about 26,000 years relative to the stars, and 20,000 years relative to its orbit pericenter. Its obliquity cycles are about 40,000 years long, and its eccentricity cycles about 100,000 and 400,000 years long.

Astronomical Solutions for Martian Paleoclimates - its obliquity varies much more, some 10 degrees, and it's difficult to predict over more than about 50 million years, making its obliquity vary between about 10 and 60 degrees. That page shows several solutions with slightly different initial conditions, and that big variation is dynamical chaos. What one sees there is typical of dynamical chaos: short-term predictability and well-defined average long-term behavior, even if unpredictable in detail.

Exoplanet Milankovitch cycles? One should not be surprised.

• Exo-Milankovitch Cycles. I. Orbits and Rotation States - IOPscience
• GENERALIZED MILANKOVITCH CYCLES AND LONG-TERM CLIMATIC HABITABILITY - IOPscience
• Milankovitch cycles of terrestrial planets in binary star systems | Monthly Notices of the Royal Astronomical Society | Oxford Academic

 

[lpetrich]

As I'd mentioned, the orbit elements' equations of motion will simplify if one can integrate over orbits. But if there are orbital resonances, this step can give bad results, so one has to track them.

 Orbital resonance - some Solar-System planets and moons are in them, with small-integer ratios of orbit periods to good approximation.

Numerous exoplanets have been discovered to be in orbital resonances, notably the seven known planets of TRAPPIST-1.

What	Ang Vel	Period	Chain
Jupiter: Io-Europa-Ganymede	4:2:1	1:2:4	2:1, 2:1
Saturn: Mimas-Tethys	2:1	1:2
Saturn: Enceladus-Dione	2:1	1:2
Saturn: Titan-Hyperion	4:3	3:4
Neptune-Pluto	3:2	2:3
TRAPPIST-1 planets	64:40:24:18:12:9:6	24:15:9:6:4:3:2	8:5, 5:3, 3:2, 3:2, 4:3, 3:2

Those three moons of Jupiter are in a "Laplace resonance":

L(Io) - 3*L(Europa) + 2*L(Ganymede) = 180d

for orbital longitudes L.

Some exoplanets are also in Laplace resonances, notably the TRAPPIST-1 planets. Going outward, each set of neighbors is in these resonances:

• 2, -5, 3
• 1, -3, 2
• 2, -5, 3
• 1, -3, 2
• 1, -2, 1

 

[lpetrich]

There are not only orbital resonances, but also spin-orbit resonances. The best-known one is the Moon's 1:1 one, from our seeing only one side of the Moon from our homeworld. Nearly all of the larger and closer moons in the Solar System are also in that state. This is a result of tidal drag, because a celestial body's primary's gravity tries to distort it into an American-football shape, and if the body rotates relative to its primary, then that would-be distortion moves through its bulk.

There are some exceptions to this rule for spin-orbit resonances. One of them is Mercury, long thought to be in a 1:1 resonance, but discovered half a century ago to be in a 3:2 resonance itself, rotating relative to the stars 3 times for every 2 orbits it makes around the Sun. This makes 1 Mercurian solar day 2 Mercurian years.

Relative to Mercury's surface, the Sun makes an approximately nephroid-shaped orbit, an orbit that looks like two circles squashed together with their touching part deleted. When the planet is the closest to the Sun, the Sun has roughly constant direction relative to the planet's surface. That is why the planet's rotation is locked in that resonance, instead of in a 1:1 resonance.

Another exception is Hyperion, with rotates chaotically with a timescale comparable to its orbit period of 21.276 (Earth) days. It has a 4:3 orbital resonance with Titan, and because of that resonance, it has a forced eccentricity of 0.123.

Hyperion has a shape that is roughly a triaxial ellipsoid, like an American football squashed along one of its short directions. Its dimensions: 360.2 km × 266.0 km × 205.4 km That shape gives Saturn's gravity some sizable "handles" to "grab".

Tidal locking and one side facing its star is expected to be common for surface-liquid-water Earth-sized planets of stars less massive than about spectral type K5 on the main sequence, mass 0.69 Msun, radius 0.74 Rsun, luminosity 0.16 Lsun, temperature 4,410 K (Sun: 5,780 K) -  Main sequence

Though exoplanets and exomoons likely often have 1:1 rotational resonances, some may instead different ratios or even chaotic rotation.

 

[lpetrich]

Orbital resonances have been used to measure the masses of some planets' moons.

1953MNRAS.113...81J Page 81

adsabs.harvard.edu

Title: On the masses of Saturn's satellites
Authors: Jeffreys, H., Sir
Journal: Monthly Notices of the Royal Astronomical Society, Vol. 113, p.81
Bibliographic Code: 1953MNRAS.113...81J

https://adsabs.harvard.edu/pdf/1955PASJ....7..176K

On the Motion of the Inner Satellites of Saturn - Y Kozai

https://www.jstage.jst.go.jp/article/pjab1945/31/6/31_6_341/_pdf

A Note on the Motion of Saturn's Satellites Enceladus and Dione - Y Kozai


The masses of Mimas, Enceladus, Tethys, Dione, and Titan have all been measured with resonances.

Mimas and Tethys, with periods 0.942 and 1.888 days, have a 2:1 resonance with "librations" or oscillations in their mean longitudes with amplitudes 44d and 5d in opposite phase and period 71 years.

Enceladus and Dione, with periods 1.370 and 2.737 days, have a 2:1 resonance, and Dione makes a forced eccentricity of Enceladus of 0.0045. Enceladus has mean-longitude librations with sizes 14m and periods 11y and 4y, while Dione has them in opposite phase with sizes 0.9m.

For Titan, one uses Hyperion's forced eccentricity, though that is rather difficult.

 

[lpetrich]

What can one do for exoplanets? Most measurements of them are not very precise, with one exception: transit times.

[2010.01074] Refining the transit timing and photometric analysis of TRAPPIST-1: Masses, radii, densities, dynamics, and ephemerides

The innermost planet has a across-self transit time of 2.889 minutes, and its orbit parameters have a time uncertainty of 15 seconds to 1 minute. The comparable across-self number for the Earth around the Sun is 7.139 minutes.

But even then, perturbation effects may be hard to observe. For an inner perturber,

\( \displaystyle{ \frac{m_P}{M} \left( \frac{a_P}{a} \right)^2 } \)

For an outer perturber,

\( \displaystyle{ \frac{m_P}{M} \left( \frac{a}{a_P} \right)^3 } \)

where M is the mass of the star, m the mass of a planet, the a's are mean distances from the star, and the P means the perturber. Also using the approximation that (smaller distance) << (larger distance).

Doing some rather hand-waving approximations in the case of close orbits, I find these approximations that should also be good for distant orbits.

\( \displaystyle{ \frac{m_P}{M} \left( \frac{a_P}{a-a_P} \right)^2 } \)

\( \displaystyle{ \frac{m_P}{M} \frac{a}{a_P} \left( \frac{a}{a_P-a} \right)^2 } \)

The planets with the largest effects on the Earth: Venus: 1.67*10^-5, Jupiter 1.04*10^-5

But those effects are very small. Doing similar calculations for the TRAPPIST-1 system gives around 10^-4.

 

[lpetrich]

Resonant effects have periods much longer than the orbit periods, and resonant effects are cumulative over those periods. That makes for much larger effects over those periods.

Longer? In the Solar System, Mimas-Tethys: 1.6 yr (free) 71 y4, Enceladus-Dione: 4 yr (free) 11 years, Titan-Hyperion: 19 years (free) 360 yr.

The "free" refers to the mean-longitude libration period. For Hyperion, https://adsabs.harvard.edu/pdf/1922BAN.....1..176W - amplitude 9.2d

These orbit details took a lot of digging up over at scholar.google.com