Astronomy

Calculating the radial tidal amplitude on a planet from the fluid Love numbers

Calculating the radial tidal amplitude on a planet from the fluid Love numbers


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How do you calculate tidal amplitudes from the fluid Love numbers? In my course on planetary physics I saw an approximate expression for the displacement from equilibrium tide:

$$Big|frac{V_T}{g}Big| approx frac{3}{4}frac{M_odot R_p^4}{M_p a^3}$$

where $V_T$ is the tide generating potential, $g$ is the gravity, $M_p$ is the mass of the planet, $R_p$ the radius and $a$ is the semi-major axis of the orbit. The displacement would then be $xi = hfrac{V_T}{g}$. This does not make a distinction between radial amplitudes at the equator or at the poles. I'm asking because I was reading this article. I was hoping the radial amplitude would correspond to my formula above (given to me without reference or derivation), but with a Love number h = 0.77 I get $xi = 0.64$m instead of $xi = 1.93$m.

The tidal potential in my notes is $$V_T(r) = -frac{GM}{a}Big(frac{r}{a}Big)^2(1 + 3ecos(M))Big[-frac{1}{2}P_2^0(cos heta) + frac{1}{4}P_2^2(cos heta)(cos(2lambda) + 4esin(M)sin(2lambda)Big] $$ $e$ is the eccentricity, the $P$ are Legendre polynomials: $P_2^0 = frac{1}{2}(3cos^2 heta - 1)$ and $P_2^2 = 3sin^3 heta$. $lambda$ is the longitude measured with respect to the long axis of the planet. I played around with this formula a bit, but I never find an amplitude of 1.93. It could be that this is just not what is meant by radial "amplitude".

The fact that no method of calculation is mentioned in the article makes me think it's either really obvious or really well known.


Retrieval of the Fluid Love Number k2 in Exoplanetary Transit Curves

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1 Deutsches Zentrum für Luft- und Raumfahrt Rutherfordstraße 2, D-12489 Berlin, Germany [email protected]

2 Technische Universität Berlin Straße des 17. Juni 135, D-10623 Berlin, Germany

3 Freie Universität Berlin Kaiserswertherstraße 16-18, D-14195 Berlin, Germany

Received 2018 December 17
Revised 2019 May 7
Accepted 2019 May 7
Published 2019 June 20

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Plain Language Summary

Due to the proximity to the Sun, tides are raised on Mercury in a similar way the Moon causes ocean tides on Earth. Although Mercury's surface is rigid, a large fluid core causes a tidal wave propagating around the planet. Using results from NASA's MESSENGER mission, we calculate that the surface should also deform by around 20 cm to 2.40 m during each Mercury orbit around the Sun. This is an amplitude that could be detected with a laser altimeter, one of the instruments onboard the upcoming BepiColombo mission. We show how the interior structure of the planet, in particular the size of an inner solid core, can be constrained by the tidal measurement. This result will help us to better understand Mercury's evolution and to constrain models explaining the magnetic field generation in Mercury's iron core.


1. Introduction

[2] Mercury has global-scale shape, gravity and tectonic characteristics which are presumably related to its formation and early evolution, and which remain poorly understood. Now that the MESSENGER spacecraft is taking new data [ Solomon et al., 2008 ], it is appropriate to reevaluate what constraints these global-scale features place on Mercury's history. In doing so, we will build on analyses based on Mariner 10 observations (detailed below) as well as more recent theoretical developments concerning the global gravity and tendency to reorient of tidally and rotationally deformed bodies [e.g., Garrick-Bethell et al., 2006 Matsuyama and Nimmo, 2007 , 2008 ]. We will conclude that Mercury's gravity field and many of its tectonic features were generated during an early epoch when its rotation rate was declining from an initially rapid rate, and which involved a true polar wander event driven by the formation of a mass excess associated with the Caloris Basin.

[3] The rest of this paper is organized as follows. The remainder of section 1 will summarize relevant work in the field, while section 2 will examine how changes in orbital eccentricity, semimajor axis, planetary rotation rate and pole orientation affect the global-scale shape, gravity and stress patterns. Section 3 discusses how the observed gravity and tectonic features constrain which of these effects is likely to have played a role. Section 4 discusses the implications of our favored scenario (a high initial spin rate and subsequent reorientation event) and makes predictions which may be tested with MESSENGER observations. Finally, details of our calculations are presented in the appendices.

[4] Mercury's present-day 3:2 spin-orbit resonance presumably occurred as the initial spin rate of the planet decreased because of solar tides [e.g., Peale, 1988 ], although how capture into this particular resonance occurred depends on the poorly understood nature of dissipation within Mercury. Dissipation within Mercury is unlikely to have been sufficient to have greatly reduced its eccentricity (current e = 0.205), which is forced to high values by secular perturbations [ Murray and Dermott, 1999 ]. Dissipation within the Sun is equally unlikely to have significantly affected its semimajor axis.

[5] Mariner 10 flybys provided constraints on the degree-2 gravity coefficients of the planet [ Anderson et al., 1987 ]. Radar ranging provided global shape measurements around the equator [ Anderson et al., 1996 ], and a single MESSENGER altimetry profile yielded an identical equatorial ellipticity, within error [ Zuber et al., 2008 ]. The equatorial ellipticity was interpreted together with the equatorial degree-2 gravity coefficient to infer a thick (100–300 km) crust [ Anderson et al., 1996 ]. Estimates of crustal thickness from relaxation studies [e.g., Watters et al., 2005 Nimmo and Watters, 2004 ] yield values at the lower end of this range.

[6] Mercury exhibits a global pattern of tectonic features [ Melosh and McKinnon, 1988 Watters and Nimmo, 2008 ]. Most of these features (lobate scarps and wrinkle ridges) are thought to be compressional in origin, equivalent to a radial contraction of <1 km [ Watters et al., 1998 ], though the nonuniform orientation and distribution of these features may point to regional-scale or nonisotropic stresses being important [ Watters et al., 2004 ]. Contraction is expected to arise as a result of core solidification [ Solomon, 1976 ], while despinning and reorientation may also have played a role [ Melosh and McKinnon, 1988 Watters and Nimmo, 2008 ].

[7] The most dramatic single feature on Mercury is the Caloris impact basin [e.g., Murchie et al., 2008 ]. There are at least two ways in which such an impact basin could generate tectonic features. First, its formation (perhaps combined with the emplacement of surrounding volcanic plains) and its subsequent infilling and relaxation could generate local tectonic features [e.g., Melosh and McKinnon, 1988 Watters et al., 2005 Kennedy et al., 2008 ]. Second, unless Caloris were exactly isostatically compensated, it would have caused planetary reorientation [e.g., Melosh and Dzusirin, 1978a Willemann, 1984 ], which in turn generates global tectonic stresses [ Melosh, 1980 ]. It is generally assumed that Caloris represents a mass excess, similar to the mass concentrations on the Moon [ Muller and Sjogren, 1968 ]. However, the location of the load could be either within the basin, and responsible for the compressional features there [ Melosh and McKinnon, 1988 ], or it could be in an annulus of volcanic plains emplaced around the basin [ Melosh and Dzurisin, 1978b ]. Melosh and Dzurisin [1978a] and Willemann [1984] used the location of Caloris and the degree-2 gravity coefficients to infer lower and upper bounds on the uncompensated annular fill thickness of 0.4 and 5 km, respectively. We treat the issue of Caloris loading and reorientation in some detail below.

[8] The amount of reorientation depends on the compensation state, and thus the effective elastic thickness Te of the feature in question. Direct measurements of Te for Mercury are currently unavailable. On the basis of inferred fault depths, Nimmo and Watters [2004] obtained Te values of 25–30 km at the time of lobate scarp formation. If Caloris is surrounded by ∼1 km of uncompensated material, then the elastic thickness must have been ∼100 km at the time of loading [ Watters and Nimmo, 2008 ]. A similar conclusion was reached by Melosh and McKinnon [1988] on the basis of the presence of extension within Caloris, and equatorial thrust faults. It seems likely that either spatial or temporal variations in Te are being recorded.


3. Dynamical Tides in a Gas Giant Planet

Following our simplified model of dynamical tides, we calculate Δk2 in a coreless, chemically homogeneous, and adiabatic Jupiter-like model. The thermal state becomes almost adiabatic in a convecting fluid planet with homogeneous composition. From the point of view of tidal calculations, the deviation from adiabaticity is negligible in the interior because the superadiabaticity required to sustain convection is a tiny fraction of the adiabatic temperature gradient, despite the possible inhibitions arising from rotation and convection. A fluid parcel in an adiabatic interior that is adiabatically displaced by a tidal perturbation will find itself in a new state that is essentially unchanged in density and temperature from the unperturbed state at that pressure. This definition of neutral stability begins to break down near the photosphere, where the density is low and the radiative time constant is no longer huge for blobs with spatial dimension of order the scale height. However, that region represents only a tiny fraction of the planet and does not produce enough gravity to significantly alter the real part of the Love number k. We discuss hypothetical contributions to k from a core and depth-varying chemical composition in Section 4.

As a matter of simplifying the arguments presented in this section, we mostly concentrate on the Love number at = m = 2, commonly known as k2. Correspondingly, k2 is forced by the degree-2 component in the gravitational pull:

Dynamical effects scale with the satellite-dependent ω. We concentrate on the dynamical effects caused by Io, the Galilean satellite with the dominant gravitational pull on Jupiter.

3.1. A Nonrotating Gas Giant

To an excellent approximation, dynamical tides in a nonrotating planet represent the forced response of the planet in the fundamental normal mode of oscillation (f-mode Vorontsov et al. 1984). Despite Cassini suggesting that higher-order normal-mode overtones (p-modes) dominate the gravitational field of Saturn's free-oscillating normal modes (Markham et al. 2020), the forced response of normal modes depends on the coupling of the gravitational pull and the mode radial eigenfunction. The forced response of the p-modes contributes negligibly to the gravitational field of dynamical tides (Vorontsov et al. 1984) because of the bad coupling between the zero-node radial component of the gravitational pull and the p-modes' eigenfunctions, the latter having one or more radial nodes. Conversely, the gravitational pull more efficiently excites f-modes, whose radial eigenfunctions roughly follow the radial scaling of the gravitational pull (∝ r ).

3.1.1. The Harmonic Oscillator Analogy

In the following, we use the forced harmonic oscillator as an analog model to tidally forced f-modes. In this model, the fractional dynamical correction to k2 acquires a simple analytical form. The equation of motion of a mass M connected in harmonic motion to a spring of stiffness and negligible dissipation is

where ω is the forcing frequency and F T is the tidal forcing. The f-modes oscillate at frequencies ω0 that are much higher than the forcing tidal frequency, meaning that tidal resonances with f-modes are highly unlikely. Assuming that the dynamical effects are small so that the tidal forcing is mostly balanced by static effects (i.e., ), the displacement of the mass is

The mass assumes the static equilibrium position us as the forcing frequency tends to zero. The displacement u is analogous to the Love number k thus, the fractional dynamical correction becomes

3.1.2. The Coriolis-free n = 1 Polytrope

To verify the analogy of the forced harmonic oscillator to tidally forced f-modes, we calculate the tidal response of a nonrotating n = 1 polytrope directly from the governing equations of tides. When Ω = 0, the governing Equation (14) reduces to

For the potential ψ at = m = 2, the boundary condition at the outer boundary r = Rp (Equation (17)) is

For the same degree and order, the continuity of the gravitational potential and its gradient at the outer boundary requires

At the center of the planet r = r0 → 0, we find the following scaling: ∇ 2 0

0, and . A finite potential ψ satisfying Equation (23) is ψ2

Similarly, a finite gravitational potential of the dynamical tides is at the center of the planet, satisfying

We compute the fractional dynamical correction to k2 by first projecting the tidal equations into spherical harmonics (Appendix B) and later solving for the relevant potentials using a Chebyshev pseudospectral numerical method (Appendix C). After projecting Equations (15) and (23) into spherical harmonics, we obtain two decoupled equations for the radial parts of the potentials ψ and (Appendix B.1). After numerically solving the radial equations in Appendix B.1 using Io's gravitational pull (ωs ≈ 42 μHz) the fractional dynamical correction corresponds to Δk2 ≈ 1.2%, in close agreement with the forced harmonic oscillator analogy applied to the oscillation frequency of the degree-2 f-mode ω0 ≈ 740 μHz (Vorontsov et al. 1976). We observe a similar agreement between the harmonic oscillator and the Coriolis-free n = 1 polytrope at higher-degree spherical harmonics (Table 1). Our results agree with a previously reported fractional correction to the gravitational coefficient C2,2k2 due to dynamical tides in a nonrotating Jupiter (Vorontsov et al. 1984).

Table 1. Io-induced Fractional Dynamical Correction Δk in a Coriolis-free Jupiter

Harmonic Oscillator n = 1 Polytrope
Type(%)(%)
(1)(2)(3)
Δk2+15+13
Δk42+5+5
Δk31+2+2
Δk33+19+15
Δk44+25+19

Note. (2) See Equation (22). The mode frequency without rotation comes from Vorontsov et al. (1976).

When Vorontsov et al. (1984) excluded Jupiter's spin, they were doing something that was mathematically sensible but physically peculiar. Tides occur much more frequently in Jupiter's rotating frame of reference, and the tidal flow is accordingly much larger than if one had Jupiter at rest, which implies a much larger dynamical effect. Consequently, Δk2 increases by an order of magnitude after partially including Jupiter's rotation in the tidally forced response of f-modes. Without the Coriolis effect but including Jupiter's spin rate (Ω ≈ 176 μHz) in the calculation of Io's tidal frequency (ω ≈ 270 μHz), the fractional dynamical correction in an n = 1 polytrope corresponds to Δk2 ≈ 13%, close to the Δk2 ≈ 15% from the forced harmonic oscillator analogy (Equation (22)). In general, for a nonrotating planet, the dynamical correction increases as the tidal frequency approaches the characteristic frequency of Jupiter's f-modes ( μHz).

3.2. The Coriolis Effect in a Rotating Gas Giant

The Galilean satellites produce dynamical tides for which the Coriolis effect plays an important role. Following relatively slow orbits (ωs Ω), the Galilean satellites produce tides on Jupiter with a tidal frequency ω

2Ω. Consequently, the two inertial terms responsible for dynamical tides on the left-hand side of the equation of motion (Equation (4)) have similar amplitudes. Moreover, Juno observes k2 to be less than the predicted number for a purely hydrostatic tide (Section 2.1), and yet our analysis above produces a positive Δk2 when dynamical effects are included and the Coriolis effect is neglected (see Equation (22)). We must accordingly motivate the change in sign when Coriolis is included.

In the following, we first calculate the gravitational effect of dynamical tides in a uniform-density sphere to reveal the fundamental behavior of the tidal equations, avoiding most of the technical difficulties related to using an n = 1 polytrope. We later found that the more complicated case of an n = 1 polytrope introduces a minor quantitative difference but leads to the same general behavior.

3.2.1. A Uniform-density Sphere

First, we explain why Δk2 changes sign because of the Coriolis effect in a specially simple model with uniform density. We calculate the fractional dynamical correction to k2 in two steps: (1) we calculate the potential of the flow ψ in a uniform-density sphere, and (2) we use the ψ calculated this way to calculate the gravity potential .

In a uniform-density sphere, the sound speed cs is infinite, and Equation (10) reduces to the well-known Poincaré problem (Greenspan et al. 1968),

where the boundary condition at the outer boundary is required to satisfy Equation (18).

Following the incompressibility of a uniform-density sphere, ψ2 retains the symmetry and degree-2 angular structure from the gravitational pull in Equation (19), thus acquiring exact solutions in the form

The numerical factor in ψ2 is set by the outer boundary condition (Equation (17)), corresponding to (Goodman & Lackner 2009)

In a constant-density sphere, tides act by displacing the sphere's boundary within an infinitesimally thin shell. According to the momentum equation, the tidal gravitational potential relates to the potential ψ following

The radial tidal displacement projected into spherical harmonics is

and the pressure perturbation follows

The gravitational potential of a thin spherical density perturbation follows directly from the definition of the gravitational potential and integration throughout the volume:

The degree-2 tidal gravitational potential corresponds to

We once again use perturbation theory to split the hydrostatic and dynamic contributions to the tidal displacement (i.e., ξ2 = ξ 0 + ξ dyn ). We first solve the well-known problem of the hydrostatic k2 (i.e., ψ = 0) in a uniform-density sphere (Love 1909). At the sphere's boundary (i.e., r = Rp), the hydrostatic gravitational potential follows 0 = 3g ξ 0 /5. From Equation (31) evaluated at r = Rp, the potential of the gravitational pull becomes T = 2g ξ 0 /5. Following the last two results, the Love number is k2 = 3/2, as expected.

The dynamical contribution to the tidal displacement ξ dyn produces the gravitational potential dyn = 3g ξ dyn /5. After applying perturbation theory and canceling the hydrostatic terms in Equation (31), the potential ψ2 becomes ψ2 = −2g ξ dyn /5. Combined with Equation (30), the last result for ψ2 allows us to reach an expression for the fractional dynamical correction in a uniform-density sphere:

Two effects contribute to the fractional dynamical correction: a negative contribution from the Coriolis effect and a positive contribution from the dynamical amplification of f-modes . The two contributions cancel each other at 2Ω = ω, where the tide achieves hydrostatic equilibrium. Tides become hydrostatic not only when the planetary spin is phase-locked with the orbit of the satellite (Ω = ωs) but also in planet–satellite systems where the central body is rotating at a rate many orders of magnitude faster than the orbit of the satellite (i.e., Ω ωs). As the frequency of the degree-2 f-mode approximately follows , Δk2 in Equation (36) approximately becomes the positive fractional correction determined in Section 3.1 after setting Ω = 0.

At the degree-2 Io-induced tidal frequency, the fractional dynamical correction corresponds to Δk2 ≈ −7.8%. The other Galilean satellites lead to a smaller Δk2 because their tidal frequency falls closer to hydrostatic equilibrium (Figure 2). A negative Δk2 works in the direction required by the nonhydrostatic component identified by Juno in Jupiter's gravity field (Section 2.1).

Figure 2. Fractional dynamical correction Δk2 in a rotating uniform-density sphere including the Coriolis effect as a function of tidal frequency (see Equation (36)).

The direction of the flow provides an explanation for the negative sign of the fractional dynamical correction via the Coriolis acceleration. By definition, a uniform-density sphere has no density perturbations in its interior and thus produces an interior tidal gravitational potential that satisfies . We adopt Equation (30) as the degree-2 potential ψ and obtain analytical solutions for the Cartesian components of the resulting degree-2 tidal flow using Equation (9),

where A is a constant depending on ξ (Appendix D). The degree-2 tidal flow purely exist in equatorial planes, showing no vertical component of motion (Figure 3(b)).

Figure 3. Degree-2 ( = m = 2) tidal perturbations on a uniform-density sphere forced by the gravitational pull of a companion satellite: (a) the nonrotating f-mode acceleration −ω 2 ξ, (b) tidal flow as shown in Equation (37), and (c) Coriolis acceleration Ω × v according to the right-hand rule.

The Coriolis acceleration plays a major role in setting the sign of the fractional dynamical correction for the Galilean satellites. Without Coriolis, the acceleration of nonrotating f-modes sustains a positive dynamical tidal displacement that follows ξ dyn ≈ 5Rp ω 2 ξ 0 /4g. A ξ dyn > 0 increases the tidal gravitational field, which leads to a positive Δk2. Conversely, as shown in Equation (36), the fractional dynamical correction flips sign when Coriolis promotes ξ dyn < 0. A Coriolis term enters the momentum equation, introducing an acceleration that competes with the acceleration of nonrotating f-modes, ultimately impacting ξ dyn . According to the right-hand rule, the Coriolis acceleration (i.e., Ω × v Figure 3(c)) opposes the direction of the acceleration of nonrotating f-modes (i.e., −ω 2 ξ Figure 3(a)). The resulting gravitational field is smaller than the hydrostatic field if ω < 2Ω, where the Coriolis acceleration beats the acceleration of nonrotating f-modes.

3.2.2. The n = 1 Polytrope

In the following, we consider the more relevant case of a compressible planet that follows an n = 1 polytropic equation of state (Equation (14)). In contrast to the localized tidal perturbation of a uniform-density sphere, a compressible body yields a tidally induced density anomaly that arises from advection of the isodensity surfaces within the body. The resulting tidal gravitational potential is different in each case owing to differences in the tidally perturbed density distribution obtained in a uniform-density sphere and a compressible body.

Despite the aforementioned difference between models, the tidal flow remains similar so that dynamical tides motivate a negative correction to k2 in each case. In an n = 1 polytrope, the continuity Equation (5) tells us that the degree-2 radial component of the flow takes the form vrj2(kr)/j1(kr) when the flow has small divergence, as it does. Remarkably, the dominant contribution to the Taylor series expansion of vr is linear in r, even out to a large fraction of the planetary radius. In a uniform-density sphere, the potential ψ is ψ2r 2 , which leads to a tidal flow that follows v2 ∝ ∇ψ2 therefore, vr is also linear in r in this model. As shown, the dominant contribution to vr scales with radius as ∝r, both in an n = 1 polytrope and in a uniform-density sphere. In an n = 1 polytrope, the dominant contribution to vr is curl- and divergence-free and provides the ψ2r 2 part of the solution to the potential ψ (Figure 4(a)). Since the n = 1 polytrope also contains terms where ψ is of higher order in r, it produces a flow with nonzero curl and divergence, causing ψ2 to depart from ψ2r 2 . Because high-order terms in r are smaller than the dominant term, dynamical effects on k2 in a uniform-density sphere are qualitatively similar to those in an n = 1 polytrope.

Figure 4. Radial functions in an n = 1 polytrope (thick blue and orange curves) of the (a) potential ψ and (b) dynamical gravitational potential dyn . The thinner black curves in panel (a) represent the radial scaling of the potential ψ in a uniform-density sphere.

We compute the fractional dynamical correction to k2 in a rotating polytrope following the same strategy used in Section 3.1.2. In opposition to the Coriolis-free polytrope, solving Equation (14) is technically challenging due to the -coupling of the potential ψ,m (e.g., mode mixing) promoted by the Coriolis effect. Mode mixing is also found in hydrostatic tides over a planet distorted by the effect of the centrifugal force (Wahl et al. 2017a). The result of projecting Equation (14) into spherical harmonics is an infinite -coupled set of ordinary differential equations for ψ,m (Appendix B.2), similarly observed in the problem of dissipative dynamical tides (Ogilvie & Lin 2004). The Coriolis-promoted -coupling comes from the sine and cosine in the spin rate of the planet ( ), which changes the degree of the spherical harmonics related to ψ. As a consequence, a given spherical harmonic from 0 on the right-hand side of Equation (14) forces multiple spherical harmonics of the potential ψ with different .

Projected into spherical coordinates, the boundary condition (Equation (17)) at r = Rp corresponds to

The outer boundary condition is also -coupled after being projected into spherical harmonics (Appendix B.2).

At the center of the planet r = r0 → 0, we find the following scaling: ∇ 2 0

0 and . As a result, the tidal Equation (14) becomes the previously solved problem of the potential ψ in a uniform-density sphere (Equation (28)) near the center. Required to be finite near the center and satisfy Equation (28), the radial part of the potential ψ follows ψ,m

r . The boundary condition for ψ,m near the center corresponds to

The equation for the gravitational potential of dynamical tides dyn remains unchanged compared to the Coriolis-free polytrope (Appendix B.1). The outer and inner boundary conditions for the gravitational potential generalize in degree as

By projecting ψ,m and ,m into a series of N Chebyshev polynomials oriented in the radial component (Appendix C), we numerically solve Equations (B4) and (B14) by truncating the infinite series of -coupled equations at an arbitrary . We choose a truncation limit and the number of Chebyshev polynomials based on numerical evidence of convergence for k2 and k42.

We obtain Δk2 = −4.0% at the degree-2 Io-induced tidal frequency (Table 2), which is of slightly lower amplitude than the estimate in a uniform-density sphere and in agreement with the k2 nonhydrostatic component observed by Juno at PJ17. Both the uniform-density sphere and the polytrope models produce fractional dynamical corrections that fall within the order-of-magnitude estimate . As argued before, the dominant contribution to the potential ψ follows the radial scaling ψr (Figure 4(a)). Ignoring the sign, the radial scaling of the dynamical gravitational potential dyn (Figure 4(b)) closely follows the shape of the hydrostatic gravitational potential (Figure 1). Due to the essentially circular and equatorial geometry of the Galilean orbits, the spherical harmonic = m = 2 dominates Jupiter's tidal gravitational field. Consequently, we concentrate on comparing the k2 Juno observation to our model prediction. Of significantly higher uncertainty, the mid-mission Juno report of Love numbers at PJ17 includes other spherical harmonics in addition to k2 (Table 2). Our polytropic model predicts an Io-induced tidal gravitational field in a 3σ agreement with most Love numbers observed at PJ17, save for k42 and k31.

Table 2. Jupiter Love Numbers

HydrostaticJuno PJ17 3σ3σ Fractional DifferenceΔk (Rotating n = 1 Polytrope)
TypeNumberNumber(%)(%)
IoEuropaGanymedeCallisto
(1)(2)(3)(4)(5)(6)(7)(8)
k20.5900.565 ± 0.018−7/−14−2−1−1
k421.7431.289 ± 0.189−37/−15 +7+8+10+12
k310.1900.248 ± 0.046+6/+55 +1+3+4+5
k330.2390.340 ± 0.116−6/+91 +2+5+7+8
k440.1350.546 ± 0.406+4/+605 +7+11+13+15

Note. (2) The hydrostatic number is from Wahl et al. (2020). (3) The Juno PJ17 3σ number is the satellite-independent number from Durante et al. (2020). (4) The 3σ fractional difference represents the minimal/maximal 3σ nonhydrostatic fractional correction required to explain the Juno observations. The fractional dynamical correction in columns (5)–(8) is valid for an n = 1 polytrope forced by the gravitational pull of the Galilean satellites. The bold values represent the fractional dynamical correction at the tidal frequency of Io, the satellite with the dominant gravitational pull on Jupiter.

3.2.3. Detection of Dynamical Tides in Systems Other than Jupiter–Io

A detection of dynamical tides via direct measurement of the gravitational field will be challenging in bodies other than Jupiter (Figure 5). The 1σ uncertainty in the gravitational field of degree-2 Io tides is projected to be σJ

6 × 10 −2 m 2 s −2 at the end of the proposed Juno extended mission (W. Folkner 2021, personal communication, 2020 April 8). The uncertainty in the measured tidal gravity field depends on the number and design of spacecraft orbits, the uncertainty in ephemerides, and instrumental capabilities. Assuming the uncertainty σJ, we roughly estimate the gravitational pull required to produce a detectable dynamical component in the gravity field using

Our calculation indicates that detecting dynamical tides in Saturn will require a mission with a more precise determination of the gravity field than that obtained by Juno (Figure 5). A 1σ detection of Europa-induced dynamical tides seems plausible at the end of Juno's extended mission, assuming that the factors determining the uncertainty in the gravity field remain similar to those of Io. We calculate a model prediction for the satellite-dependent Jupiter Love number for all of the Galilean satellites (Table 2). We obtain k2 = 0.578 in the case of Europa, a prediction testable by the recently approved Juno extended mission.

Figure 5. Conditions for the detection of dynamical tides evaluated for the Galilean satellites (black) and inner Saturn satellites (white). Satellites to the right of the dashed line have favorable conditions for a detection of dynamical tides assuming an uncertainty roughly similar to that of Io's k2 on Jupiter at the end of Juno's extended mission. The fractional dynamical correction Δk2 is for an n = 1 polytrope.


4. The Period and Q of the Chandler Wobble

[29] Sea surface height observations from the T/P altimeter are derived after applying environmental corrections to the raw altimeter range measurements for the range delays that are caused by the wet and dry troposphere, the ionosphere, and the sea state bias. Geophysical corrections for the mean sea surface, the inverse barometer response, and the lunisolar solid Earth, ocean, and load tides, are applied to the sea surface height measurements. The diurnal and semidiurnal ocean and load tides from the GOT99.2b model [ Ray, 1999 ] and an equilibrium model of the long-period lunisolar ocean tides are adopted for the ocean and load tide corrections. The sea surface height observations are not corrected in any way for the pole tide.

[30] Attitude control system errors may have degraded the quality of the T/P measurements during the first nine repeat cycles of the mission, so only data from repeat cycles 10 to 331, corresponding to 21 December 1992 to 18 September 2001, are considered. A repeat cycle refers to the fact that the T/P ground track exactly repeats once every 9.9156 days, and might be considered to be the sampling interval of the global sea surface heights. Data from the experimental single frequency solid state altimeter on the T/P satellite are also excluded from this analysis since they do not have the same measurement accuracies as the data from the dual frequency altimeter [ Fu et al., 1994 ]. The T/P satellite has an orbit with an inclination angle of 66° so sea surface height measurements are only available within the latitudes of ±66°. Long gaps in the time series of T/P sea surface height measurements are introduced at polar latitudes because the oceans in those regions are covered by ice for a significant portion of each year. Such long data gaps are likely to corrupt any estimates of sea surface height variations at the Chandler wobble period of 433 days, so the T/P data used here are further limited to latitudes within ±60°.

[31] There are various sources of seasonal variations in the sea surface heights and these are indistinguishable from those associated with the pole tide. Therefore, least squares estimates of the annual and semiannual variations are removed from both the altimetric sea surface height observations, and the locations of the rotation pole prior to correlating the two types of observations. Furthermore, least squares estimates of mean and secular variations are also always removed from both the altimetric and rotation pole data, and should accommodate any errors in the assumed drift of the mean rotation pole. Residual variations in the rotation pole and residual pole tide deformations of the sea surface are then expected to occur almost completely at the Chandler wobble period.

[34] Increasing the assumed value of h2 by 1% increases the estimated value of k2 by less than 0.15%. Note also that if the second term of equation (22) is ignored then the estimated amplitude and phase of k2 is 0.363 and −4.6°. This illustrates the importance of accounting for the self-gravitation and loading of the ocean pole tide since this difference of 0.055, or 18%, in the estimated amplitude of k2 is mostly explained by the ignored term γ2(1 + k2′)α2a21 = 0.047. It is also worth noting that if the reported pole was assumed to be identical to the rotation pole, that is m(t) = p(t), instead of using equation (7) to relate the reported pole to the rotation pole, then the estimated value of k2 at the Chandler wobble period would be larger by approximately 0.003, or 1%.

[35] An attempt is also made to observe any shorter wavelength features of the ocean pole tide. Maps of the geocentric pole tide admittance function, as defined by ZA(θ, λ) in equation (21), are estimated simultaneously with the mean, drift, and seasonal response in the 3 by 3° geographic bins. As before, the admittance functions are with respect to the rotation pole data that have seasonal variations removed. These estimates of the admittance are smoothed with a Gaussian smoothing function similar to that used by Desai and Wahr [1995] . Figure 5 shows these maps together with the predicted equilibrium geocentric admittance function as defined by A(θ, λ) in equation (22) using values of h2 = 0.6027 and k2 = 0.308.

[36] These maps confirm the long-wavelength agreement between the observed and predicted geocentric equilibrium response. There are apparent short-wavelength departures from the expected equilibrium response. However, many of these short-wavelength features are correlated with strong oceanographic phenomenon. As such, they are probably better interpreted as noise in the estimated geocentric pole tide admittance function that is being introduced by the inadequate decoupling of the general ocean circulation of the oceans from the geocentric pole tide deformation from the limited 9 year data record. For example, the observed real admittance function shows features in the equatorial regions of the Pacific Ocean that are highly correlated with the strong El Niño event that occurred in 1997–1998. Sea surface heights as large as 25 cm are associated with this event and are much larger than the 9 cm global standard deviation of the sea surface height anomaly that is typically observed by T/P. Another example is in the observed imaginary admittance where there is an apparent departure from equilibrium that corresponds to the Gulf Stream on the eastern coast of North America.

[37] To place these results in perspective note that the expected maximum admittance of approximately 50% corresponds to a displacement of the ocean surface of 8–18 mm over the time period of sea surface height observations that are used in this analysis. These amplitudes are no larger than 8% of the amplitudes of the sea surface heights that are associated with the 1997–1998 El Niño event. Also, the maps of the geocentric pole tide admittance function are generated with only 7.5 cycles of the Chandler wobble.

[39] Both of the uncertainties quoted above result from the formal errors in the respective estimates of k2. The apparent reduction in these uncertainties by an order of magnitude when using the 1 Hz data instead of the spherical harmonic approach is a numerical artifact that can be explained by the fact that in each repeat cycle a 3 by 3 degree bin contains an average of 150 1 Hz measurements. Meanwhile, the spherical harmonic approach effectively includes only a single measurement in each 3 by 3 degree bin once every repeat cycle. As such, the formal errors are expected to be smaller by a factor of (150) 1/2 ≈ 12 when using the 1 Hz data. Since the 1 Hz altimetric measurements have accuracies of 40–50 mm then the averaged once per repeat cycle measurements in each 3 by 3 degree bin might be considered to have accuracies of 3–4 mm. The formal errors that are determined from the spherical harmonic approach are therefore likely to provide a more appropriate measure of the uncertainty in the estimated value of k2 than do the formal errors that result from the use of the 1 Hz data. Alternatively, the formal errors from the use of the 1 Hz data should be amplified by at least an order of magnitude to account for the errors in each 1 Hz measurement.

[40] Figure 6 illustrates the impact that additional T/P observations of the degree 2 order 1 spherical harmonic components of the sea surface heights have on the estimated values and associated formal errors of the amplitude and phase of k2. The values and formal errors shown in Figure 6 are computed by incrementally including an additional pair of degree 2 order 1 spherical harmonic coefficients into the estimate of k2 as they have become available from each T/P repeat cycle during the course of the T/P mission. No significant reduction in the formal errors is likely from any additional T/P data. However, the Chandler wobble amplitude is time variable and only those amplitudes between 1992 and 2001, which ranged from 0.12 to 0.26 arcseconds, are considered in this figure. The sea surface height observations would have had increased sensitivity to the pole tide if the Chandler wobble amplitudes had remained at their maximum values throughout the observation period, and the formal errors would have been smaller but by no more than a factor of 2.

[41] The scatter of the amplitude and phase that are estimated as each of the last 131 available repeat cycles, or approximately 3 Chandler periods, are incrementally included into the estimate of k2 is 0.013 and 5.5°, respectively. These values of the scatter of the estimates of k2 are smaller than the respective uncertainties quoted earlier when all of the available spherical harmonic observations are used, particularly for the amplitude where the scatter is smaller by a factor of more than 2. This suggests that the quoted uncertainties are somewhat conservative. While additional T/P data are unlikely to significantly reduce the formal errors, these longer durations of observations should reduce the scatter in the estimated values of the amplitude and phase of k2.

[42] Table 1 compares the amplitude and phase of k2 that is inferred from polar motion observations of the period and Q of the Chandler wobble with the result that is determined here from T/P altimetry. The inferred values of k2 are derived by assuming a self-consistent equilibrium pole tide response with the amplitude of h2 = 0.6027, the phase of h2 constrained to be identical to that of k2, and a real load Love number k2′ = −0.3075 [ Farrell, 1972 ]. Most notable is the fact that the uncertainties in the value of k2 that is estimated from T/P altimetry is one order of magnitude larger than that inferred from observations of the period and Q of the Chandler wobble. Mantle anelasticity causes the solid Earth to lag the elastic response such that the phase lag ϵ is expected to be positive [ Wahr, 1985 ], and this is confirmed by observations of the Chandler wobble Q. The phase of k2 determined here from T/P altimetry is negative but fortunately the associated uncertainty in the phase encompasses the expected positive value.

Author ϵ, deg
Jeffreys [1968] a a Values of the Love number k2 are inferred from observations of the Chandler wobble period and Q by assuming a self-consistent equilibrium ocean pole tide response, with the amplitude of h2 = 0.6027, the phase of h2 constrained to be identical to that of k2, and a real load Love number k2′ = −0.3075.
0.307 ± 0.003 0.9 + 0.6/−0.6
Currie [1974] a a Values of the Love number k2 are inferred from observations of the Chandler wobble period and Q by assuming a self-consistent equilibrium ocean pole tide response, with the amplitude of h2 = 0.6027, the phase of h2 constrained to be identical to that of k2, and a real load Love number k2′ = −0.3075.
0.307 ± 0.001 0.8 + 0.3/−0.2
Wilson and Haubrich [1976] a a Values of the Love number k2 are inferred from observations of the Chandler wobble period and Q by assuming a self-consistent equilibrium ocean pole tide response, with the amplitude of h2 = 0.6027, the phase of h2 constrained to be identical to that of k2, and a real load Love number k2′ = −0.3075.
0.308 ± 0.003 0.6 + 0.6/−0.4
Ooe [1978] a a Values of the Love number k2 are inferred from observations of the Chandler wobble period and Q by assuming a self-consistent equilibrium ocean pole tide response, with the amplitude of h2 = 0.6027, the phase of h2 constrained to be identical to that of k2, and a real load Love number k2′ = −0.3075.
0.309 ± 0.003 0.6 + 0.6/−0.4
This paper 0.308 ± 0.035 −5.1 ± 7.1
  • a Values of the Love number k2 are inferred from observations of the Chandler wobble period and Q by assuming a self-consistent equilibrium ocean pole tide response, with the amplitude of h2 = 0.6027, the phase of h2 constrained to be identical to that of k2, and a real load Love number k2′ = −0.3075.

4. Discussion

KELT-24 b has some key characteristics that make it a compelling target for detailed characterization. Specifically, the host star is very bright, V = 8.3 mag, and the planet is quite massive, . With such a high mass, it is interesting to see some signs that it is inflated (RP = 1.272 ± 0.021 ). However, this is not unique to this system since many massive hot Jupiters have inflated radii. Of all the hot Jupiters known, KELT-24 b is one of only a few dozen massive (MP = 4–13 ) hot Jupiters (P < 10 days) with a host star bright enough (V < 13 mag) to permit detailed characterization. 61 At V = 8.3 mag, KELT-24 is the brightest known planetary host in this regime (see Figure 8). The host star, KELT-24, has a mass of M = , a radius of R = 1.506 ± 0.022 , and an age of Gyr. It is the brightest star known to host a transiting giant planet with a period between 5 and 10 days, and one of the longest period planets discovered from ground-based surveys. Interestingly, HAT-P-2b (Bakos et al. 2007) is quite similar to KELT-24 b in that they have almost the same orbital period (5.63 days compared to 5.55 days), similar planetary masses (9.0 MJ compared to 5.2 MJ), and both host stars that are very bright (HAT-P-2 is V = 8.7 mag). The relatively young age of KELT-24 suggests it has just started to evolve from the zero-age main sequence, which is consistent with our UVW analysis (see Section 2.7).

Figure 8. Distribution of planet mass and orbital period for the known population of radial velocity only (gray) and transiting hot Jupiters (colored by optical magnitude). The size of the circle is scaled by the host star's apparent brightness. The filled-in circle represents the location of KELT-24 b. We only show systems that have a 3σ or better measurement on the planet's mass. The horizontal dashed line is the lower limit (4 ) of the massive hot Jupiters regime we discuss in Section 4. The data behind this figure was downloaded from the composite table on UT 2019 May 07 from the NASA Exoplanet Archive (Akeson et al. 2013).

We detected a nonzero, small 3σ eccentricity of for KELT-24 b's orbit. However, systems observed to have small eccentricities (<0.1) are subject to the Lucy–Sweeney bias, where observational errors of a circular orbit can lead to the detection of a slight eccentricity (Lucy & Sweeney 1971). Therefore, we caution the reader about the detection of the eccentricity, even though it is detected at a formally significant confidence level. We do note that this eccentricity was not only constrained by the spectroscopic observations from TRES (see Section 2.3) but also from the KELT-FUN transit observations (i.e., the transit duration), because they are all globally modeled with EXOFASTv2 (see Section 3). Because the eccentricity is quite small and not conclusive, we use Equation (3) from Adams & Laughlin (2006) to approximate the circularization timescale of KELT-24 b to be 12.7 Gyr (assuming Q = 10 6 ). This circularization timescale does not change significantly when accounting for the small eccentricity detected. Since the age of KELT-24 is significantly smaller than the circularization timescale, we do not assume the eccentricity to be zero within our global analysis. Future observations should confirm this nonzero eccentricity by obtaining additional higher precision RVs and/or observing the secondary eclipse of KELT-24 b. The time difference between the secondary eclipse assuming zero eccentricity and one using e = 0.078 from our results is about 3.5 hr. Future eclipse observations should account for this when scheduling eclipse observations. KELT-24 has a projected rotational velocity of 19.46 ± 0.18 km s −1 , corresponding to a rotation period of 3.9 days. Since this is shorter than the orbital period of KELT-24 b we do not expect the planet to be tidally synchronized.

4.1. Tidal Evolution and Irradiation History

We calculated the past and future orbital evolution of the orbit of KELT-24 b under the influence of tides, using the POET code (Penev et al. 2014). We calculated the evolution of the orbital semimajor axis (see Figure 9) under the assumptions of a constant tidal phase lag (or constant tidal quality factor), circular orbit, and no perturbations due to further, undetected, objects in the system. Under these assumptions, the tides that the star raises on the planet have no appreciable effect on the orbit, because the angular momentum that can be stored/extracted from the planet is a negligible fraction of the total orbital angular momentum. As a result, the tidal evolution is dominated by the dissipation of tidal perturbations in the star. We accounted for the evolution of the stellar radius, assuming a MIST (Choi et al. 2016 Dotter 2016) stellar evolutionary track appropriate for the best-fit stellar mass and metallicity from our global fit (see Section 3). Finally, we combined the evolution of the orbital semimajor axis with the evolution of the stellar luminosity per the same MIST model to calculate the evolution of the amount of irradiation received by the planet (see Figure 9). Because the tidal dissipation in stars is poorly constrained, and likely not well described by a simple constant phase lag model, we considered a broad range of plausible phase lags, parameterized by the commonly used tidal dissipation parameter (the ratio of the tidal quality factor Q and the Love number, k2).

Figure 9. Evolution of the semimajor axis (top) and irradiation (bottom) for KELT-24 b shown for a range of values for Q . The color of the line indicates the dissipation in the star (green: Q = 10 5 , lavender: Q = 10 7 , gold: Q = 10 8 ).

Regardless of the tidal quality factor, we concluded that the planet has always been subject to a level of irradiation several times larger than the 2 × 10 8 erg s −1 cm −2 threshold Demory & Seager (2011) suggest is required for the planet to be significantly inflated. Also, again regardless of the amount of dissipation, the planet has undergone at most moderate orbital evolution prior to its current, nearly circular orbit. In contrast, the future fate of the planet is significantly impacted by the amount of tidal dissipation assumed. For a tidal quality factor of , the planet will be engulfed by its parent star within a few hundred Myr, while for or larger the planet survives until the end of the main sequence life of its parent star.

4.2. KELT-24's Aligned Orbit

KELT-24 b's aligned orbit is interesting in the context of its mass, possible small eccentricity, and the young age of the system. Hébrard et al. (2010) noted that for massive hot Jupiters, their orbits are typically prograde but with a nonzero misalignment angle, a pattern that still holds true today (see Figure 10). KELT-24 b is therefore somewhat unusual in that its sky-projected spin–orbit misalignment λ is consistent with zero, although the true 3D spin–orbit misalignment ψ could be larger if the host star is not viewed equator-on. We cannot measure the inclination of the stellar rotation axis I using our current data, but a TESS measurement of the rotation period via spot modulation or asteroseismology could allow this measurement.

Figure 10. Spin–orbit misalignment of KELT-24b in context with the population of hot Jupiters from the literature. We show the sky-projected spin–orbit misalignments as a function of planetary mass. Red and blue plot points denote planets orbiting stars with effective temperatures less than or greater than the Kraft break at 6250 K, respectively hot Jupiters orbiting cooler stars typically have well-aligned orbits, whereas those orbiting hotter stars like KELT-24 have a wide range of misalignments (Winn 2010). We highlight KELT-24b as the large dark red star the uncertainties are smaller than the plot symbol size. We show only planets with measured masses rather than upper limits, and uncertainties on the spin–orbit misalignments of less than 20°.

Furthermore, KELT-24's young age and slightly eccentric, aligned orbit place some constraints upon the past history of the system. Some of the high-eccentricity migration mechanisms, such as the Kozai–Lidov mechanism (Anderson et al. 2016) or secular planet–planet interactions (Petrovich & Tremaine 2016) may take hundreds of Myr and typically leave planets in highly eccentric, misaligned orbits. Together with the long tidal damping timescale for the system (longer than the age of the universe), this suggests that KELT-24 b likely instead migrated through a faster, less dynamically violent mechanism such as interactions with the protoplanetary disk or in situ formation.

4.3. Atmospheric Characterization Prospects

As mentioned, KELT-24 b is one of the few known massive giant planets orbiting a host star bright enough to allow for detailed atmospheric characterization observations. The other comparable planets in this mass range that have been observed with either Spitzer or HST are HAT-P-2 b (Spitzer, Lewis et al. 2014), WASP-14 b (Spitzer, Wong et al. 2015), Kepler-13A b (Spitzer and HST, Beatty et al. 2017), KELT-1b (Spitzer, Beatty et al. 2019), and WASP-103b (Spitzer and HST, Kreidberg et al. 2018). Interestingly, KELT-24 b orbits the brightest host in this regime and has the lowest blackbody equilibrium temperature of all these planets: approximately 1450 K. This places KELT-24 b in a different and potentially interesting atmospheric regime. Given the similarities between HAT-P-2b and KELT-24, future atmospheric observation KELT-24 b would provide a nice comparison to those already taken for HAT-P-2b.

Observations of massive field brown dwarfs have shown that there is a strong blueward shift in the NIR colors of these objects as they cool from roughly 1400 K down to approximately 1000 K. This is known as the "L–T" transition, and is generally believed to represent the clouds in the atmospheres of the hotter L-dwarfs slowly dropping below the level of the photosphere in the cooler T-dwarfs. The few observations we have of giant exoplanets in this regime indicate that this transition may occur at cooler temperatures, presumably because the lower surface gravity of the planets is altering the cloud dynamics in their atmospheres, perhaps allowing vertical lofting to maintain the clouds higher for longer (Triaud et al. 2015).

KELT-24 b possesses an intermediate surface gravity, 3 times higher than Jupiter but 10 times lower than a brown dwarf, that straddles previous observations. The characterization of the global cloud properties on KELT-24 b therefore could allow us to better understand the dynamical processes behind the L–T transition. In particular, a recent analysis of Spitzer phase-curve results by Beatty et al. (2019) has shown that all hot Jupiters appear to posses a nightside cloud deck at a temperature of roughly 1000 K. The relatively low equilibrium temperature of KELT-24 b's atmosphere indicates that even dayside clouds on KELT-24 b would be close in composition to the universal nightside clouds on other hot Jupiters. The spectroscopic measurement of KELT-24 b's emission might, therefore, be able to determine the specific composition of these clouds. Cloud compositions would in turn provide invaluable insight into the cloud condensation processes, and hence dynamics.


Contents

Italian scientist Giovanni Battista Riccioli and his assistant Francesco Maria Grimaldi described the effect in connection with artillery in the 1651 Almagestum Novum, writing that rotation of the Earth should cause a cannonball fired to the north to deflect to the east. [12] In 1674 Claude François Milliet Dechales described in his Cursus seu Mundus Mathematicus how the rotation of the Earth should cause a deflection in the trajectories of both falling bodies and projectiles aimed toward one of the planet's poles. Riccioli, Grimaldi, and Dechales all described the effect as part of an argument against the heliocentric system of Copernicus. In other words, they argued that the Earth's rotation should create the effect, and so failure to detect the effect was evidence for an immobile Earth. [13] The Coriolis acceleration equation was derived by Euler in 1749, [14] [15] and the effect was described in the tidal equations of Pierre-Simon Laplace in 1778. [16]

Gaspard-Gustave Coriolis published a paper in 1835 on the energy yield of machines with rotating parts, such as waterwheels. [17] That paper considered the supplementary forces that are detected in a rotating frame of reference. Coriolis divided these supplementary forces into two categories. The second category contained a force that arises from the cross product of the angular velocity of a coordinate system and the projection of a particle's velocity into a plane perpendicular to the system's axis of rotation. Coriolis referred to this force as the "compound centrifugal force" due to its analogies with the centrifugal force already considered in category one. [18] [19] The effect was known in the early 20th century as the "acceleration of Coriolis", [20] and by 1920 as "Coriolis force". [21]

In 1856, William Ferrel proposed the existence of a circulation cell in the mid-latitudes with air being deflected by the Coriolis force to create the prevailing westerly winds. [22]

The understanding of the kinematics of how exactly the rotation of the Earth affects airflow was partial at first. [23] Late in the 19th century, the full extent of the large scale interaction of pressure-gradient force and deflecting force that in the end causes air masses to move along isobars was understood. [24]

In Newtonian mechanics, the equation of motion for an object in an inertial reference frame is

Transforming this equation to a reference frame rotating about a fixed axis through the origin with angular velocity ω >> having variable rotation rate, the equation takes the form

The fictitious forces as they are perceived in the rotating frame act as additional forces that contribute to the apparent acceleration just like the real external forces. [25] [26] The fictitious force terms of the equation are, reading from left to right: [27]

As the Coriolis force is proportional to a cross product of two vectors, it is perpendicular to both vectors, in this case the object's velocity and the frame's rotation vector. It therefore follows that:

  • if the velocity is parallel to the rotation axis, the Coriolis force is zero. (For example, on Earth, this situation occurs for a body on the equator moving north or south relative to Earth's surface.)
  • if the velocity is straight inward to the axis, the Coriolis force is in the direction of local rotation. (For example, on Earth, this situation occurs for a body on the equator falling downward, as in the Dechales illustration above, where the falling ball travels further to the east than does the tower.)
  • if the velocity is straight outward from the axis, the Coriolis force is against the direction of local rotation. (In the tower example, a ball launched upward would move toward the west.)
  • if the velocity is in the direction of rotation, the Coriolis force is outward from the axis. (For example, on Earth, this situation occurs for a body on the equator moving east relative to Earth's surface. It would move upward as seen by an observer on the surface. This effect (see Eötvös effect below) was discussed by Galileo Galilei in 1632 and by Riccioli in 1651. [29] )
  • if the velocity is against the direction of rotation, the Coriolis force is inward to the axis. (On Earth, this situation occurs for a body on the equator moving west, which would deflect downward as seen by an observer.)

The time, space and velocity scales are important in determining the importance of the Coriolis force. Whether rotation is important in a system can be determined by its Rossby number, which is the ratio of the velocity, U, of a system to the product of the Coriolis parameter, f = 2 ω sin ⁡ φ , and the length scale, L, of the motion:

The Rossby number is the ratio of inertial to Coriolis forces. A small Rossby number indicates a system is strongly affected by Coriolis forces, and a large Rossby number indicates a system in which inertial forces dominate. For example, in tornadoes, the Rossby number is large, in low-pressure systems it is low, and in oceanic systems it is around 1. As a result, in tornadoes the Coriolis force is negligible, and balance is between pressure and centrifugal forces. In low-pressure systems, centrifugal force is negligible and balance is between Coriolis and pressure forces. In the oceans all three forces are comparable. [30]

An atmospheric system moving at U = 10 m/s (22 mph) occupying a spatial distance of L = 1,000 km (621 mi), has a Rossby number of approximately 0.1.

A baseball pitcher may throw the ball at U = 45 m/s (100 mph) for a distance of L = 18.3 m (60 ft). The Rossby number in this case would be 32,000.

Baseball players don't care about which hemisphere they're playing in. However, an unguided missile obeys exactly the same physics as a baseball, but can travel far enough and be in the air long enough to experience the effect of Coriolis force. Long-range shells in the Northern Hemisphere landed close to, but to the right of, where they were aimed until this was noted. (Those fired in the Southern Hemisphere landed to the left.) In fact, it was this effect that first got the attention of Coriolis himself. [31] [32] [33]

Tossed ball on a rotating carousel Edit

The figure illustrates a ball tossed from 12:00 o'clock toward the center of a counter-clockwise rotating carousel. On the left, the ball is seen by a stationary observer above the carousel, and the ball travels in a straight line to the center, while the ball-thrower rotates counter-clockwise with the carousel. On the right the ball is seen by an observer rotating with the carousel, so the ball-thrower appears to stay at 12:00 o'clock. The figure shows how the trajectory of the ball as seen by the rotating observer can be constructed.

On the left, two arrows locate the ball relative to the ball-thrower. One of these arrows is from the thrower to the center of the carousel (providing the ball-thrower's line of sight), and the other points from the center of the carousel to the ball. (This arrow gets shorter as the ball approaches the center.) A shifted version of the two arrows is shown dotted.

On the right is shown this same dotted pair of arrows, but now the pair are rigidly rotated so the arrow corresponding to the line of sight of the ball-thrower toward the center of the carousel is aligned with 12:00 o'clock. The other arrow of the pair locates the ball relative to the center of the carousel, providing the position of the ball as seen by the rotating observer. By following this procedure for several positions, the trajectory in the rotating frame of reference is established as shown by the curved path in the right-hand panel.

The ball travels in the air, and there is no net force upon it. To the stationary observer, the ball follows a straight-line path, so there is no problem squaring this trajectory with zero net force. However, the rotating observer sees a curved path. Kinematics insists that a force (pushing to the right of the instantaneous direction of travel for a counter-clockwise rotation) must be present to cause this curvature, so the rotating observer is forced to invoke a combination of centrifugal and Coriolis forces to provide the net force required to cause the curved trajectory.

Bounced ball Edit

The figure describes a more complex situation where the tossed ball on a turntable bounces off the edge of the carousel and then returns to the tosser, who catches the ball. The effect of Coriolis force on its trajectory is shown again as seen by two observers: an observer (referred to as the "camera") that rotates with the carousel, and an inertial observer. The figure shows a bird's-eye view based upon the same ball speed on forward and return paths. Within each circle, plotted dots show the same time points. In the left panel, from the camera's viewpoint at the center of rotation, the tosser (smiley face) and the rail both are at fixed locations, and the ball makes a very considerable arc on its travel toward the rail, and takes a more direct route on the way back. From the ball tosser's viewpoint, the ball seems to return more quickly than it went (because the tosser is rotating toward the ball on the return flight).

On the carousel, instead of tossing the ball straight at a rail to bounce back, the tosser must throw the ball toward the right of the target and the ball then seems to the camera to bear continuously to the left of its direction of travel to hit the rail (left because the carousel is turning clockwise). The ball appears to bear to the left from direction of travel on both inward and return trajectories. The curved path demands this observer to recognize a leftward net force on the ball. (This force is "fictitious" because it disappears for a stationary observer, as is discussed shortly.) For some angles of launch, a path has portions where the trajectory is approximately radial, and Coriolis force is primarily responsible for the apparent deflection of the ball (centrifugal force is radial from the center of rotation, and causes little deflection on these segments). When a path curves away from radial, however, centrifugal force contributes significantly to deflection.

The ball's path through the air is straight when viewed by observers standing on the ground (right panel). In the right panel (stationary observer), the ball tosser (smiley face) is at 12 o'clock and the rail the ball bounces from is at position 1. From the inertial viewer's standpoint, positions 1, 2, and 3 are occupied in sequence. At position 2 the ball strikes the rail, and at position 3 the ball returns to the tosser. Straight-line paths are followed because the ball is in free flight, so this observer requires that no net force is applied.

The force affecting the motion of air "sliding" over the Earth's surface is the horizontal component of the Coriolis term

This component is orthogonal to the velocity over the Earth surface and is given by the expression

In the northern hemisphere where the sign is positive this force/acceleration, as viewed from above, is to the right of the direction of motion, in the southern hemisphere where the sign is negative this force/acceleration is to the left of the direction of motion

Rotating sphere Edit

Consider a location with latitude φ on a sphere that is rotating around the north–south axis. [34] A local coordinate system is set up with the x axis horizontally due east, the y axis horizontally due north and the z axis vertically upwards. The rotation vector, velocity of movement and Coriolis acceleration expressed in this local coordinate system (listing components in the order east (e), north (n) and upward (u)) are:

When considering atmospheric or oceanic dynamics, the vertical velocity is small, and the vertical component of the Coriolis acceleration is small compared with the acceleration due to gravity. For such cases, only the horizontal (east and north) components matter. The restriction of the above to the horizontal plane is (setting vu = 0):

By setting vn = 0, it can be seen immediately that (for positive φ and ω) a movement due east results in an acceleration due south. Similarly, setting ve = 0, it is seen that a movement due north results in an acceleration due east. In general, observed horizontally, looking along the direction of the movement causing the acceleration, the acceleration always is turned 90° to the right and of the same size regardless of the horizontal orientation.

As a different case, consider equatorial motion setting φ = 0°. In this case, Ω is parallel to the north or n-axis, and:

Accordingly, an eastward motion (that is, in the same direction as the rotation of the sphere) provides an upward acceleration known as the Eötvös effect, and an upward motion produces an acceleration due west.


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Acknowledgments

The bottom-pressure-record data used in this study are available as supporting information to Ray [ 2013 ] at http://onlinelibrary.wiley.com/doi/10.1002/jgrc.20336/suppinfo. SDD performed the work described in this paper at the Jet Propulsion Laboratory, California Institute of Technology under contract with the National Aeronautics and Space Administration. RDR was supported by the National Aeronautics and Space Administration's Ocean Surface Topography project.

The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.