# No theoretical basis for an orbit-spin coupling inertial force in planetary atmospheres

Thomas Pierron, François Forget

► To cite this version:

Thomas Pierron, François Forget. No theoretical basis for an orbit-spin coupling inertial force in planetary atmospheres. Icarus, 2026, 450, pp.116984. 10.1016/j.icarus.2026.116984. hal-05505534

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Research Paper

# No theoretical basis for an orbit-spin coupling inertial force in planetary atmospheres

T. Pierron ID *, F. Forget

Laboratoire de Météorologie Dynamique, IPSL, Sorbonne Université, CNRS, École Polytechnique, ENS, Paris, France

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

Mars atmosphere

Planetary dynamics

Celestial Mechanics

Orbital dynamics

Atmospheric circulation

## ABSTRACT

Correlations between Martian orbital dynamics and the interannual variability of global dust storms (GDS) were reported by Shirley (2015) and Shirley and Mischna (2017). To explain this correlation, Shirley (2017) proposed a theoretical derivation of an inertial force acting on the Martian atmosphere, depending on both the planet's spin rate and its orbital angular momentum. This term, referred to as the “coupling term acceleration” (CTA), has been hypothesized to provide a dynamical forcing mechanism for GDS initiation. In this study, we revisit the orbit-spin coupling hypothesis by re-deriving the inertial forces acting on a parcel of air in the atmosphere of a rotating, orbiting planet. We show that no orbit-spin coupling term naturally appears in the equations of motion. Furthermore, the inertial accelerations associated with the planet's orbital motion are almost entirely compensated by the external gravitational field, leaving only a well-known gravitational tidal term. This result demonstrates that there is no theoretical basis for a direct orbit-spin coupling force acting on planetary atmospheres.

## 1. Introduction

Variability in planetary atmospheres can have multiple origins. It is often difficult to fully understand it due to the chaotic nature of meteorology (Lorenz, 1963). In the case of Mars, a remarkable interannual variability is observed in the occurrence of dust storms. In some years, extreme events occur in which atmospheric dust encircles the entire planet for several months, forming what are called global dust storms (GDS). These GDS are intense meteorological phenomena caused by widespread lifting of dust from the Martian surface. Although regional dust storms occur every year during southern spring and summer, only some years experience planet-encircling storms that produce a dramatic increase in dust opacity and obscure the surface from orbit (Zurek and Martin, 1993; Kahre et al., 2017). The physical mechanisms underlying this interannual variability remain poorly understood.

Several studies (e.g. Mulholland et al., 2013; Newman and Richardson, 2015) have attempted to reproduce the observed dust cycle using numerical global climate models (GCMs). These works suggest that the availability of surface dust may be a controlling factor. After a GDS, the surface dust reservoir is depleted, allowing only regional storms for a few subsequent years until the supply is replenished. However, such models have not yet succeeded in fully explaining the observed timing and occurrence of GDS.

In this context, Shirley (2015) proposed an alternative explanation for the interannual variability of GDS occurrence. He identified a

statistical correlation between the timing of global storms and the phase (as defined by Shirley, 2015) of Mars' orbital angular momentum, measured with respect to the solar system barycenter. Building on this empirical correlation, Shirley (2017) developed a theoretical framework arguing that an inertial force, coupling Mars' spin to its orbital motion, should act on the atmosphere and modulate the likelihood of GDS initiation.

This hypothesis was partly inspired by earlier debates on possible planetary influences on solar activity, in which tidal interactions or orbital configurations were hypothesized to affect the solar dynamo. Although such solar-planetary coupling hypotheses have been extensively discussed, recent studies have shown that they are physically implausible, primarily because of the extremely small amplitude of planetary tides in the solar interior (Nataf, 2022; Cionco et al., 2023). Nevertheless, the broader exploration of external dynamical forcings in astrophysical and planetary contexts motivated the consideration of analogous orbit-spin coupling mechanisms at the planetary scale.

In Shirley (2017), an hypothesized force, referred to as the Coupling Term Acceleration (CTA), depends explicitly on the time derivative of Mars' orbital angular momentum. Several subsequent studies (e.g. Mischna and Shirley, 2017; Newman et al., 2019) have attempted to implement the CTA in GCMs and assess its impact on Martian meteorology, but results remain inconclusive.

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: thomas.pierron@lmd.ipsl.fr (T. Pierron).</footnote>
More recently, Shirley et al. (2020) extended the historical record of Martian GDS and argued that including the CTA term improves hind-casting and even forecasting of GDS occurrences. They noted, however, that orbital angular momentum variability is unlikely to be the sole trigger, and that other key factors, notably surface dust availability, must play a central role.

The works of Shirley (2015, 2017), Shirley et al. (2020) have led to significant discussion about possible connections between Mars' orbital dynamics and GDS initiation, motivating a careful reassessment of the underlying theoretical framework. In this paper, we first re-examine the theoretical basis of the CTA as derived in Shirley (2017). When the acceleration of a fluid parcel is correctly derived in a rotating and orbiting reference frame, no term coupling the spin and orbital motions appears in the momentum equation. Only the classical inertial terms, Coriolis, centrifugal, and Euler, are recovered.

We then extend the derivation to include the planet's orbital motion, and we identify the additional inertial terms: an orbital centrifugal term, an orbital Coriolis term, and an orbital Euler term proportional to the time derivative of the orbital angular velocity. These terms are real inertial forces acting on atmospheric parcels. However, when the fundamental principle of dynamics is applied to the entire planet-atmosphere system, these inertial forces are almost entirely balanced by the external gravitational field, leaving only the well-known tidal force. This tidal term can indeed couple the planet's spin to its orbit, but only indirectly and over very long timescales, with negligible impact on short-term atmospheric variability such as GDS occurrence.

In other words, while a true orbit-spin coupling does exist through gravitational tides, it does not provide a direct or rapidly varying forcing capable of driving short-term atmospheric variability, such as the initiation of global dust storms. Consequently, the CTA term as implemented in previous studies lacks a reliable theoretical basis and should not be included in the atmospheric momentum budget.

**Notation.** Bold symbols denote three-dimensional vectors, while nonbold symbols denote scalar quantities. A superimposed dot indicates the time derivative of a scalar quantity. For vector quantities, time derivatives are written using the operator $\frac{d}{dt}$ with an explicit reference frame subscript (e.g. $\frac{dA}{dt}|_{\mathcal{R}_0}$) when needed.

## 2. The orbit-spin coupling theory

### 2.1. Summary of the method

In Section 3 of Shirley (2017), the author derives a new inertial force. It depends on the cross product of the orbital angular velocity and the rotational angular velocity of the planet. This force is claimed to influence the motion of a particle as observed from the local rotating frame of the planet. To obtain this result, the author considers the velocity of a particle in an inertial reference frame defined by fixed points relative to distant stars. The particle velocity is then differentiated with respect to time in this inertial frame. From there, the author introduces two different rotating reference frames: the first, called the $\alpha$-frame, rotates with the planet's proper rotation at angular velocity $\omega_\alpha$; the second, called the $\beta$-frame, rotates at the orbital angular velocity $\omega_\beta$.

After expressing the velocity of a particle in the inertial frame, the author attempts to derive the inertial force acting on the particle by calculating the time derivative with respect to the $\alpha$-frame of the velocity of the particle in the $\alpha$-frame.

### 2.2. Rewriting the derivation

In this section we reproduce the derivation of Shirley (2017) using Shirley's original notation to identify precisely where the error occurs.

The inconsistency arises from an incorrect transformation of a time derivative between rotating reference frames, which leads to the appearance of an artificial coupling term that depends on the arbitrary choice of an intermediate frame.

The starting point (Eq. 4 of Shirley, 2017) is the expression of the velocity $V$ of an air parcel in the inertial frame:

$$V = v_i + \omega_\alpha \times r + v_\alpha \quad (1)$$

where $v_i$ is the translational velocity of the planet along its orbit and $r$ is the position vector from the planet center to the parcel, $\omega_\alpha$ and $v_\alpha$ are the proper spin angular velocity of the planet and the velocity relative to the rotating $\alpha$-frame, respectively.

Shirley explicitly writes that

> "the $v_i$ are purely translational motions; as they are traditionally considered to have no rotary component, these play no role in our subsequent development. We are thus free to discard the first term."

This statement is incorrect: although $v_i$ represents translational motion, it is neither rectilinear nor uniform, since the planet follows an accelerated elliptic orbit. Its time derivative therefore produces a non-zero acceleration, which in turn leads to inertial forces that must be included in the momentum equation.

Nevertheless, in order to make the identification of the error as transparent as possible, let us follow Shirley's assumption and omit $v_i$ for the remainder of this section (the correct treatment of the $v_i$ term, and its contribution to the inertial force, will be derived explicitly in Section 3).

The inertial velocity becomes (Eq. 5 of Shirley, 2017):

$$V = \omega_\alpha \times r + v_\alpha \quad (2)$$

Shirley then defines the angular velocity of the planet as seen from the $\beta$-frame, rotating with the orbital motion:

$$\omega_\rho = \omega_\alpha - \omega_\beta, \quad (3)$$

The inertial velocity can be rewritten as (Eq. 8 of Shirley, 2017):

$$V = \omega_\beta \times r + \omega_\rho \times r + v_\alpha. \quad (4)$$

This velocity is then differentiated with respect to time in the inertial frame (Eq. 9a of Shirley, 2017):

$$\frac{dV}{dt} = \omega_\rho \times \frac{dr}{dt} + \frac{d\omega_\beta}{dt} \times r + \omega_\rho \times \frac{dr}{dt} + \frac{d\omega_\beta}{dt} \times r + \frac{dv_\alpha}{dt} \quad (5)$$

Then, the author introduces a new notation for the time derivative of $\omega_\rho$, writing:

$$\frac{d\omega_\rho}{dt} \Big|_{\text{observed}} = \frac{d\omega_\rho}{dt} \Big|_{\text{orbital}} + \frac{d\omega_\alpha}{dt}. \quad (6)$$

The acceleration is then rewritten as (Eq. 9b of Shirley, 2017):

$$\frac{dV}{dt} = \omega_\rho \times \frac{dr}{dt} + \frac{d\omega_\beta}{dt} \times r + \omega_\rho \times \frac{dr}{dt} + \left( \frac{d\omega_\rho}{dt} \Big|_{\text{orbital}} + \frac{d\omega_\alpha}{dt} \right) \times r + \frac{dv_\alpha}{dt} \quad (7)$$

Note that Eq. 9b of Shirley (2017) as printed omits the final “$\times r$” in the fourth term, which is clearly required for dimensional consistency; here we include it for clarity.

Shirley then writes:

> "This expansion allows us to drop the 'observed' and 'orbital' subscripts in all subsequent steps, as we are now accounting for the geophysical (i.e., $d\omega_\alpha/dt$) and planetary dynamical sources (i.e., $d\omega_\rho/dt$ orbital) separately. Accordingly, $d\omega_\rho/dt$ will hereafter be identified solely and exclusively with the orbital contribution."

This choice is legitimate as a notation convention, but it must be kept in mind that $d\omega_\rho/dt$ no longer strictly denotes the time derivative of $\omega_\rho$ (which would be $d\omega_\alpha/dt - d\omega_\beta/dt$). Instead, it represents only the orbital part, i.e. $-d\omega_\beta/dt$.

Following this convention, in Shirley (2017) the acceleration becomes:

$$\frac{dV}{dt} = \omega_\rho \times \frac{dr}{dt} + \frac{d\omega_\beta}{dt} \times r + \omega_\rho \times \frac{dr}{dt} + \left( \frac{d\omega_\rho}{dt} + \frac{d\omega_\alpha}{dt} \right) \times r + \frac{dv_\alpha}{dt} \quad (8)$$
Eqs. (5) and (8) are nearly identical, except that Eq. (8) now contains the explicit term $(d\omega_\alpha/dt) \times r$.

This is not an algebraic error: it simply reflects a change in notation for $d\omega_\rho/dt$, which is no longer interpreted as the full time derivative of $\omega_\rho$.

Shirley expresses all time derivatives in the inertial frame as functions of their counterparts in the $\beta$-frame (Eqs. 11a-11f of Shirley, 2017). In particular (Eq. 11d of Shirley, 2017):

$$ \left. \frac{d\omega_\rho}{dt} \right|_{\text{inertial}} = \left. \frac{d\omega_\rho}{dt} \right|_{\beta} + \omega_\beta \times \omega_\rho \qquad (9) $$

At first sight, this Eq. 11d of Shirley (2017) appears to be directly derived from the standard transport theorem. However, because in Shirley's notation $d\omega_\rho/dt$ represents only the orbital part $-d\omega_\beta/dt$, applying the transport theorem to $-\omega_\beta$ yields:

$$ \left. \frac{d\omega_\rho}{dt} \right|_{\text{inertial}} = - \left. \frac{d\omega_\beta}{dt} \right|_{\beta} - \omega_\beta \times \omega_\beta = - \left. \frac{d\omega_\beta}{dt} \right|_{\beta}, $$

since $\omega_\beta \times \omega_\beta = 0$. Thus, no cross-product term should appear if one keeps track of Shirley's definition of $d\omega_\rho/dt$.

<table><thead><tr><th>Shirley (2017), Eq. 11d</th><th>Consistent expression<br>(using $d\omega_\rho/dt = -d\omega_\beta/dt$)</th></tr></thead><tbody><tr><td>$\left. \frac{d\omega_\rho}{dt} \right|_{\text{inertial}} = \left. \frac{d\omega_\beta}{dt} \right|_{\beta} + \omega_\beta \times \omega_\rho$</td><td>$\left. \frac{d\omega_\beta}{dt} \right|_{\text{inertial}} = \left. \frac{d\omega_\beta}{dt} \right|_{\beta}$</td></tr></tbody></table>

Then Shirley switches to the $\alpha$-frame and correctly cancels or simplifies all terms by taking into account that $\left. \frac{d\omega_\rho}{dt} \right|_{\beta} = -\left. \frac{d\omega_\beta}{dt} \right|_{\alpha}$, arriving at the final expression which is almost correct. The term introduced earlier due to his notation, $(\left. \frac{d\omega_\alpha}{dt} \right) \times r$, is properly canceled. However, the second term, which should be zero under his notation, $\omega_\beta \times \omega_\rho$, persists until the end.

This leads to the final expression of the effective force (Eq. 16 in Shirley, 2017):

$$ F_{\text{effective}} = m \frac{dv_\alpha}{dt_\alpha} = F_{\text{total}} - m\omega_\alpha \times (\omega_\alpha \times r) - 2m\omega_\alpha \times v_\alpha - m \frac{d\omega_\alpha}{dt_\alpha} \times r - m(\omega_\beta \times \omega_\rho) \times r \qquad (10) $$

The final term of this expression is presented as a new inertial force coupling the spin angular speed and the orbital angular rate, called an “orbit-spin coupling” which would affect a particle and might therefore play a role in the atmosphere of the body. As shown in the derivation above, this term $\omega_\beta \times \omega_\rho (= \omega_\beta \times \omega_\alpha)$ has no physical basis and should not appear in the momentum equation.

It is important to note that, in all previous studies addressing the orbit-spin coupling effect on the Martian atmosphere (e.g. Shirley, 2017; Shirley and Mischna, 2017; Mischna and Shirley, 2017; Newman et al., 2019; Shirley et al., 2020), the orbit-spin coupling acceleration $(\omega_\beta \times \omega_\alpha) \times r$, which we show must vanish, is nevertheless used after being modified in a way that introduces further inconsistencies.

Shirley wished to express the (nonexistent) inertial term $(\omega_\beta \times \omega_\alpha) \times r$ in terms of the orbital angular momentum of the planet L. For this purpose, the correct substitution should have been

$$ \omega_\beta = \frac{L}{MR^2}, $$

where M is the planetary mass and R its instantaneous distance to the solar system barycenter, leading to the inertial acceleration:

$$ \frac{1}{MR^2} (L \times \omega_\alpha) \times r, $$

which already has the correct dimensions of an acceleration and requires no additional factor. Moreover, L and its time derivative $\dot{L}$ are not proportional quantities, so the term cannot be expressed as a simple multiple of L.

However, Shirley (2017) replaced $\omega_\beta$ by L without the $1/(MR^2)$ factor and then took the time derivative of the entire expression, introducing $\dot{L}$ and an empirical scaling coefficient c:

$$ \text{CTA} = -c (\dot{L} \times \omega_\alpha) \times r. $$

This operation fundamentally alters the physical meaning of the term, since $(\dot{L} \times \omega_\alpha) \times r$ has the dimensions of a torque rate times a distance (i.e. kg m$^3$ s$^{-3}$), not of an acceleration.

In Shirley (2017), the use of $\dot{L}$ was primarily motivated by numerical considerations rather than physical reasons. Because L has an extremely large magnitude (~10$^{39}$ kg m$^2$ s$^{-1}$ for Mars), its time derivative was used instead and interpreted as a “tendency”, a small, time-varying quantity that could be integrated at each time step in a numerical climate model. The coefficient c was introduced as a free “coupling efficiency” parameter to rescale this tendency to realistic atmospheric wind amplitudes. The value of c (~10$^{-13}$) found empirically roughly compensates for the missing $1/(MR^2)$ factor, but has no theoretical justification.

However, in subsequent works (e.g. Mischna and Shirley, 2017; Newman et al., 2019; Shirley et al., 2020), this crucial distinction was lost. The CTA was no longer treated as a tendency on the acceleration, but directly interpreted and applied as an actual acceleration field (with units of m s$^{-2}$). This confusion further detaches the CTA from any consistent physical derivation, since it was originally introduced only as a numerical convenience, not as a physically valid inertial force.

### 3. Derivation of inertial forces in a rotating-orbiting frame

#### 3.1. Introduction

While the orbit-spin coupling term proposed by Shirley (2017) is not correct, a very weak inertial force depending on the time variation of the orbital angular momentum can still be derived. During his derivation of the inertial force, Shirley started from the particle velocity expressed as $V = \omega_\alpha \times r + v_\alpha$, dropping the translational velocity of the body along its orbit around the solar system barycenter, $v_t$ (Eqs 4 and 5 in Shirley, 2017). He argued that this velocity corresponds purely to translational motion and therefore would not affect the inertial forces. While this is true in the case of uniform rectilinear translation, it is not perfectly true for a planet in orbit because the orbital frame is not inertial: it undergoes acceleration relative to the inertial frame. Consequently, neglecting $v_t$ overlooks inertial effects due to this acceleration. In this section, we will show that by retaining the translational velocity, it is possible to derive inertial accelerations that do not introduce a coupling between orbital and spin motions, but do yield inertial forces dependent on the orbital angular momentum and its time derivative.

#### 3.2. Method

In order to derive all the inertial forces on a particle in the planet's atmosphere induced by both the orbital motion and the rotation of a planet around the barycenter of the solar system, we define three reference frames:

* $\mathcal{R}_0$, the inertial frame centered at the barycenter of the solar system and fixed with respect to the distant stars,

* $\mathcal{R}_1$, the frame centered at the barycenter of the solar system that rotates with the planet's orbital angular velocity

* $\mathcal{R}_2$, the rotating frame co-moving with the planet, i.e., both orbiting the barycenter of the solar system and rotating with the planet.

Let $\omega$ denote the rotational angular velocity of the planet in $\mathcal{R}_0$ (associated with its spin), $r$ the position vector of the studied particle with respect to the center of the planet expressed in $\mathcal{R}_2$, $v$ the velocity measured with respect to the orbiting-rotating frame $\mathcal{R}_2$ and $R$ the
<figure><img src="image_4.png" /><figcaption>Fig. 1. Schematic representation of the reference frames and vectors used in this study. $\mathcal{R}_0$ is the inertial frame fixed at the barycenter of the Solar System, $\mathcal{R}_1$ rotates with the planet's orbital angular velocity $\Omega$, and $\mathcal{R}_2$ is co-rotating with the planet's spin angular velocity $\omega$. The vectors $\mathbf{R}$, $\mathbf{r}$, and $\mathbf{v}$ represent the position of the planet, the position of a particle in its atmosphere, and its velocity in $\mathcal{R}_2$, respectively. The unit vectors $e_r$ and $e_\theta$ indicate the radial and orthoradial orbital directions relative to the barycenter.</figcaption></figure>

position vector of the planet's center of mass with respect to the solar system barycenter. The bold symbols represent three-dimensional vectors, whose orientations are illustrated in Fig. 1.

We can write the velocity of the particle measured in the inertial frame $\mathcal{R}_0$ as:

$$V = \left. \frac{d\mathbf{R}}{dt} \right|_{\mathcal{R}_0} + \omega \times \mathbf{r} + \mathbf{v} \quad (11)$$

where $\left. \frac{d}{dt} \right|_{\mathcal{R}}$ denotes the derivative with respect to time in the reference frame $\mathcal{R}$.

Although the transport theorem involved in this derivation is standard (e.g. Goldstein et al., 1950; Landau and Lifshitz, 1960), we detail the intermediate steps explicitly here because the emergence of the proposed orbit-spin coupling term arises on this specific transformation.

$$\left. \frac{d\mathbf{V}}{dt} \right|_{\mathcal{R}_0} = \left. \frac{d^2 \mathbf{R}}{dt^2} \right|_{\mathcal{R}_0} + \left. \frac{d}{dt}(\omega \times \mathbf{r}) \right|_{\mathcal{R}_0} + \left. \frac{d\mathbf{v}}{dt} \right|_{\mathcal{R}_0} \quad (12)$$

$$= \left. \frac{d^2 \mathbf{R}}{dt^2} \right|_{\mathcal{R}_0} + \left. \frac{d\omega}{dt} \right|_{\mathcal{R}_0} \times \mathbf{r} + \omega \times \left. \frac{d\mathbf{r}}{dt} \right|_{\mathcal{R}_0} + \left. \frac{d\mathbf{v}}{dt} \right|_{\mathcal{R}_0} \quad (13)$$

$$= \left. \frac{d^2 \mathbf{R}}{dt^2} \right|_{\mathcal{R}_0} + \left( \left. \frac{d\omega}{dt} \right|_{\mathcal{R}_2} + \omega \times \omega \right) \times \mathbf{r} + \omega \times \left( \left. \frac{d\mathbf{r}}{dt} \right|_{\mathcal{R}_2} + \omega \times \mathbf{r} \right) + \left. \frac{d\mathbf{v}}{dt} \right|_{\mathcal{R}_2} + \omega \times \mathbf{v} \quad (14)$$

$$= \left. \frac{d\mathbf{v}}{dt} \right|_{\mathcal{R}_2} + \omega \times (\omega \times \mathbf{r}) + 2\omega \times \mathbf{v} + \left. \frac{d\omega}{dt} \right|_{\mathcal{R}_2} \times \mathbf{r} + \left. \frac{d^2 \mathbf{R}}{dt^2} \right|_{\mathcal{R}_0} \quad (15)$$

The second, third, and fourth terms correspond respectively to the spin centrifugal, Coriolis, and Euler accelerations. The last term corresponds to the orbital acceleration of the planet's center of mass. This acceleration can be decomposed into radial and orthoradial components as:

$$\left. \frac{d^2 \mathbf{R}}{dt^2} \right|_{\mathcal{R}_0} = \left. \frac{d}{dt} \right|_{\mathcal{R}_0} (\dot{R} e_r + R \Omega e_\theta) \quad (16)$$

$$\left. \frac{d^2 \mathbf{R}}{dt^2} \right|_{\mathcal{R}_0} = (\ddot{R} - R\Omega^2)e_r + (2\dot{R}\Omega + R\dot{\Omega})e_\theta \quad (17)$$

Here, $e_r = \mathbf{R}/\|\mathbf{R}\|$ is the radial unit vector pointing from the solar system barycenter toward the planet's center of mass, and $e_\theta$ is the orthoradial unit vector defined by $e_\theta = e_z \times e_r$, where $e_z$ is the unit vector normal to the orbital plane. The scalar $\Omega$ denotes the instantaneous orbital angular velocity magnitude, and $R = \|\mathbf{R}\|$ the distance between the solar system barycenter and the planet's center

of mass. The dot notation denotes the derivative with respect to time in $\mathcal{R}_0$.

To facilitate comparison with Shirley (2017), who expressed his presumed coupling term as a function of the orbital angular momentum of the planet $L$, we also rewrite the orbital acceleration in terms of:

$$L = M R \times \left. \frac{d\mathbf{R}}{dt} \right|_{\mathcal{R}_0} \quad (18)$$

where $M$ is the planet's mass.

For planar orbital motion, $L$ is perpendicular to the orbital plane and can be written as:

$$L = M R^2 \Omega e_z \quad (19)$$

The orbital acceleration can then be expressed in terms of the orbital angular momentum as:

$$\left. \frac{d^2 \mathbf{R}}{dt^2} \right|_{\mathcal{R}_0} = \left( \ddot{R} - \frac{L^2}{M^2 R^3} \right) e_r + \frac{\dot{L}}{M R} e_\theta \quad (20)$$

where $L = \|\mathbf{L}\|$ denotes the magnitude of the orbital angular momentum.

As $\mathcal{R}_0$ is an inertial frame, we can apply the fundamental principle of dynamics to the particle in this reference frame:

$$F = m \left. \frac{d\mathbf{V}}{dt} \right|_{\mathcal{R}_0} \quad (21)$$

where $m$ is the mass of the particle and $F$ is the total true external force acting on it, including the gravitational field, pressure-gradient forces, viscous drag, and contact forces.

The forces felt by the particle in the orbiting-rotating reference frame $\mathcal{R}_2$ can then be written as the sum of the true external force and the inertial forces:

$$F_{\text{felt}} = F + F_{\text{inertial}} = m \left. \frac{d\mathbf{v}}{dt} \right|_{\mathcal{R}_2} \quad (22)$$

$$F_{\text{felt}} = F - m\omega \times (\omega \times \mathbf{r}) - 2m\omega \times \mathbf{v} - m \left. \frac{d\omega}{dt} \right|_{\mathcal{R}_2} \times \mathbf{r} - \left( m\ddot{R} - \frac{mL^2}{M^2 R^3} \right) e_r - \frac{m\dot{L}}{M R} e_\theta \quad (23)$$

The fifth term, along the direction of $e_r$, represents the difference between the acceleration in the reference frame $\mathcal{R}_1$ and the centrifugal acceleration associated with the orbital motion of the planet. The last term acts in the orthoradial direction of the orbit and depends on the
time derivative of the planet's orbital angular momentum L. In a simple two-body problem, this term vanishes since L is conserved when only a central gravitational force acts. However, in the solar system, L varies due to the gravitational perturbations of other planets, and this term becomes non-zero.

It is therefore important to highlight that there indeed exists an inertial force acting on the particle which depends both on the instantaneous orbital angular momentum, through the centrifugal term, and its time variation, through the $\dot{L}$ term. This is an authentic inertial effect arising from the non-inertial nature of the orbiting frame, but it is fundamentally different from the so-called “Coupling Term Acceleration” (CTA) introduced by Shirley (2017), as it does not involve any spin-orbit coupling.

## 4. Dynamical balance at the center of mass

In the previous section, the momentum balance was applied to a parcel of the atmosphere, revealing the presence of orbital inertial forces that depend explicitly on the acceleration of the planet's center of mass (Eq. (15)). To understand how these terms behave, we now apply the momentum balance to the whole planet at its center of mass. By comparing the two expressions, we find that the inertial forces acting on individual parcels are almost entirely compensated by the gravitational field acting on the planet itself. What remains is only the small differential term associated with the spatial variation of the external gravitational field, the well-known gravitational tides.

To simplify the expression of the forces felt by a particle in the orbiting-rotating frame $\mathcal{R}_2$ (Eq. (23)), we first apply Newton's second law to the entire body (solid planet plus atmosphere). Internal forces (pressure, viscosity, planet's self-gravity) cancel by action-reaction, so only external forces contribute to the acceleration of the center of mass (CM). Denoting by $M$ the total mass (solid + atmosphere), by $V$ the volume of the body, by $R$ the position of the center of mass (CM), and by $\rho(r)$ the mass density expressed in CM-centered coordinates $r$, we have:

$$M \frac{d^2 \mathbf{R}}{dt^2} \Big|_{\mathcal{R}_0} = \iiint_V \rho(r) g_{\text{ext}}(\mathbf{R} + r) d^3 r \quad (24)$$

where $g_{\text{ext}}$ is the external gravitational field (from the Sun and other planets, excluding the planet's self-gravity).

Expanding the external field about the CM,

$$g_{\text{ext}}(\mathbf{R} + \mathbf{r}) = g_{\text{ext}}(\mathbf{R}) + \nabla g_{\text{ext}}(\mathbf{R}) \cdot \mathbf{r} + \mathcal{O}(\|\mathbf{r}\|^2), \quad (25)$$

where $\nabla g_{\text{ext}}(\mathbf{R})$ denotes the Jacobian of the external gravitational field, evaluated at $\mathbf{R}$.

Using the CM identity $\int_V \rho \mathbf{r} dV = 0$, we obtain

$$M \frac{d^2 \mathbf{R}}{dt^2} \Big|_{\mathcal{R}_0} = M g_{\text{ext}}(\mathbf{R}) \quad (26)$$

to first order.

Let us rewrite the expression of the force felt by a particle in the planet's atmosphere, as derived in Section 3.

$$F_{\text{felt}} = F_{\text{contact}} + m g_{\text{ext}}(\mathbf{R} + \mathbf{r}) + m g_{\text{self}}(\mathbf{r}) \\ - m \omega \times (\omega \times \mathbf{r}) - 2m \omega \times \mathbf{v} - m \frac{d\omega}{dt} \Big|_{\mathcal{R}_2} \times \mathbf{r} - m \frac{d^2 \mathbf{R}}{dt^2} \Big|_{\mathcal{R}_0}. \quad (27)$$

where we replace $F$ by all true forces $F_{\text{contact}}$ including pressure gradient, viscous stresses, and any other local contact forces, and where $g_{\text{self}}(\mathbf{r})$ is the planet's self-gravity.

Using Eq. (26), we obtain:

$$F_{\text{felt}} = F_{\text{contact}} + m g_{\text{self}}(\mathbf{r}) - m \omega \times (\omega \times \mathbf{r}) - 2m \omega \times \mathbf{v} - m \frac{d\omega}{dt} \Big|_{\mathcal{R}_2} \times \mathbf{r} \\ + m[g_{\text{ext}}(\mathbf{R} + \mathbf{r}) - g_{\text{ext}}(\mathbf{R})]. \quad (28)$$

Assuming the first-order Taylor expansion of $g_{\text{ext}}$ given in Eq. (25), this becomes (see Box I).

Comparing this expression with the one derived in Section 3, we see that no inertial force depending on the orbital angular momentum or its time derivative is ultimately felt by the particle. It is crucial to emphasize that these inertial forces do exist in the equations of motion; however, when the total true force $F$ is expanded to include the external gravitational field, these inertial contributions are almost entirely compensated. What remains is simply the tidal term $m \nabla g_{\text{ext}}(\mathbf{R}) \cdot \mathbf{r}$, which represents the differential gravitational effect across the body. This shows that the variation of the orbital angular momentum does not introduce a direct forcing term on the atmosphere, contrary to Shirley's hypothesis. However, it is worth recalling that a coupling between the orbital and rotational angular momenta of a planet does exist, but it does not act as a direct force on atmospheric parcels. Instead, this coupling manifests through tidal torques that slowly modify the planet's spin rate $\omega$ and obliquity over long timescales (e.g. Murray and Dermott, 1999), as described by the angular momentum equation

$$\frac{d}{dt}(I \omega) = \int_V \rho(\mathbf{r}) \mathbf{r} \times [\nabla g_{\text{ext}}(\mathbf{R}) \cdot \mathbf{r}] d^3 \mathbf{r}, \quad (30)$$

where $I$ is the planet's moment of inertia about the spin axis. The tidal torque on the right-hand side generally describes both non-dissipative effects, such as precession and nutation, and dissipative components that can, over long timescales, drive $\omega$ toward the orbital angular velocity $\Omega$, leading to spin-orbit synchronization (or pseudo-synchronization if the orbit is eccentric).

These results demonstrate that the orbit-spin coupling does not manifest as an instantaneous inertial force acting on the atmosphere, but rather as a gradual evolution of the planet's spin rate driven by gravitational tidal torques, a process occurring on timescales much longer than those relevant for atmospheric variability.

We note that gravitational tides do produce instantaneous differential forces within the planet-atmosphere system, including semi-diurnal tidal deformations. However, these gravitational tides depend solely on the external gravitational potential and its spatial gradient, and not on the orbital angular momentum or its time variation. In planetary atmospheres, these gravitational tides are generally negligible compared to atmospheric thermal tides, which are driven by periodic stellar heating and depend primarily on star-planet distance rather than on orbital angular velocity or angular momentum (Lindzen and Chapman, 1969).

## 5. Conclusion

We showed that there is no inertial force depending simultaneously on the planetary spin rate $\omega$ and the orbital angular velocity $\Omega$ that would form a true orbit-spin coupling, thus invalidating the demonstration proposed by Shirley (2017). The so-called Coupling Term Acceleration (CTA), employed in several studies of orbit-spin coupling (Shirley, 2017; Shirley and Mischna, 2017; Mischna and Shirley, 2017; Newman et al., 2019; Shirley et al., 2020), has no solid theoretical basis and should not be included in the momentum budget of the atmosphere.

We did show, however, that some inertial forces involving the orbital angular momentum $L$ and its time derivative $\dot{L}$ do appear in the equations of motion. Yet, by applying the fundamental principle of dynamics to the whole body (solid and atmosphere) at the center of mass, we demonstrated that these inertial contributions are nearly entirely compensated by the external gravitational field. What remains is a purely gravitational tidal term, which indeed represents a genuine orbit-spin coupling but acts only indirectly, by exerting a torque that slowly modifies the planet's spin rate $\omega$ and, to a lesser extent, its orbital parameters, over timescales that are orders of magnitude longer than those relevant for atmospheric variability.

Consequently, such tidal coupling cannot explain the interannual variability of the Martian atmosphere, nor the occurrence of global dust
$$F_{\text{felt}} = F_{\text{contact}} + m g_{\text{self}}(\mathbf{r}) - m \omega \times (\omega \times \mathbf{r}) - 2m \omega \times \mathbf{v} - m \frac{d\omega}{dt} \Big|_{\mathcal{R}_2} \times \mathbf{r} + m \nabla g_{\text{ext}}(\mathbf{R}) \cdot \mathbf{r} \qquad (29)$$

## Box I.

storms. The variability of these events is much more likely linked to atmospheric dynamics and/or limited surface dust availability, as shown by Mulholland et al. (2013) and Newman and Richardson (2015).

Only in environments with very strong tidal gradients, for instance planets or moons orbiting close to massive bodies such as giant planets or compact objects, could such effects have a direct impact on the climate or atmospheric circulation.

### CRediT authorship contribution statement

T. Pierron: Writing – original draft, Methodology, Investigation, Formal analysis, Conceptualization. F. Forget: Writing – original draft, Methodology, Investigation, Formal analysis, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 835275, project “Mars Through Time”).

### Data availability

No data was used for the research described in the article.

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