# Exoplanet Nodal Precession Induced by Rapidly Rotating Stars:
Impacts on Transit Probabilities and Biases

Alexander P. Stephan Department of Astronomy, The Ohio State University,
Columbus, OH 43210, USA  Center for Cosmology and AstroParticle Physics, The
Ohio State University, Columbus, OH 43210, USA  B. Scott Gaudi Department of
Astronomy, The Ohio State University, Columbus, OH 43210, USA

(Received XXX; Revised YYY; Accepted ZZZ)

###### Abstract

For the majority of short period exoplanets transiting massive stars with
radiative envelopes, the spin angular momentum of the host star is greater
than the planetary orbital angular momentum. In this case, the orbits of the
planets will undergo nodal precession, which can significantly impact the
probability that the planets transit their parent star. In particular, for
some combinations of the spin-orbit angle $\psi$ and the inclination of the
stellar spin $i_{*}$, all such planets will eventually transit at some point
over the duration of their precession period. Thus, as the time over which the
sky has been monitored for transiting planets increases, the frequency of
planets with detectable transits will increase, potentially leading to biased
estimates of exoplanet occurrence rates, especially orbiting more massive
stars. Furthermore, due to the dependence of the precession period on orbital
parameters such as spin-orbit misalignment, the observed distributions of such
parameters may also be biased. We derive the transit probability of a given
exoplanet in the presence of nodal precession induced by a rapidly spinning
host star. We find that the effect of nodal precession has already started to
become relevant for some short-period planets, i.e., Hot Jupiters, orbiting
massive stars, by increasing transit probabilities by of order a few percent
for such systems within the original $Kepler$ field. We additionally derive
simple expressions to describe the time evolution of the impact parameter $b$
for applicable systems, which should aid in future investigations of exoplanet
nodal precession and spin-orbit alignment.

Exoplanets (498) — Hot Jupiters (753) — Transits (1711)

##  1 Introduction

Over recent decades thousands of exoplanets and exoplanet candidates have been
discovered and characterized due to the combined efforts of space-based and
ground-based observational instruments and their surveys, such as by Kepler
(Borucki et al., 2010), the Transiting Exoplanet Survey Satellite (TESS;
Ricker et al. 2015), the Hungarian Automated Telescope (HAT; Bakos et al.
2007), the Hungarian Automated Telescope-South (HATS; Bakos et al. 2013), the
Wide Angle Search for Planets (WASP; Pollacco et al. 2006), and the Kilodegree
Extremely Little Telescope (KELT; Pepper et al. 2007), among others. While our
understanding of exoplanet formation, system architectures, and evolution is
far from complete, these surveys have revealed large populations of exoplanets
that are very unlike our own solar system planets. In particular, many
exoplanets with periods much shorter than that of Mercury have been
discovered, such as Hot or Warm Jupiters, as well as previously unknown
exoplanet types such as Super-Earths and Mini-Neptunes111See
https://exoplanetarchive.ipac.caltech.edu/ for database of known exoplanets.

Interestingly, short period exoplanets have been found around many stars much
more massive than the Sun, such as A-type stars (e.g., Collier Cameron et al.
2010, Shporer et al. 2011, Szabó et al. 2012, Gaudi et al. 2017), which are
known to generally rotate rapidly for extended parts of their main sequence
lifetime due to inefficient magnetic braking and tidal dissipation of their
angular momentum (e.g., Kraft, 1967; Ward et al., 1976). Exoplanets transiting
such stars can display interesting new phenomena that are less important for
planets transiting less rapidly rotating stars, including distorted and
asymmetric transits (Barnes, 2009), which can be used to measure the spin-
orbit alignment of the planet (Barnes et al., 2011), gravity-darkened
‘seasons’ (Ahlers et al., 2020), and orbital and/or spin precession (Barnes et
al., 2013).

The nature of the precession depends on the magnitude of the angular momentum
of the stellar spin $L_{*}$ compared to the magnitude of the angular momentum
of the planetary orbit $L_{p}$ (e.g., Murray & Dermott, 2000). When $L_{*}\gg
L_{p}$, the planet orbital angular momentum vector precesses about the stellar
spin angular momentum vector. When $L_{p}\gg L_{*}$, the stellar spin angular
momentum vector precesses about the planetary orbital angular momentum vector.
When $L_{*}\sim L_{p}$ then both the planetary orbital angular moment vector
and the stellar spin angular moment vector mutually precess around the net
angular momentum vector of the system. Here we focus on the first case, where
the angular momentum of the stellar rotation dominates over the orbital
angular momentum of its planet, which leads to nodal precession of the
planet’s orbit. This is generally the case for short period planets orbiting
hot stars above the Kraft break (Kraft 1967, Ward et al. 1976, Figure 1).

In some cases, precession is observable over relatively short observation time
frames of only a few years (Johnson et al., 2015; Watanabe et al., 2020; Borsa
et al., 2021; Stephan et al., 2022). The nodal precession of an exoplanet’s
orbit due to its host star’s rapid rotation is a powerful tool to study the
structural response of stars to the deformation caused by such rotation. So
far, three rapidly precessing exoplanets have been observed and had their
precession rates and stellar gravitational quadrupole moments measured (Szabó
et al., 2012; Johnson et al., 2015; Watanabe et al., 2020; Borsa et al., 2021;
Stephan et al., 2022).

Beyond studying stellar structure, nodal precession also has an impact on our
ability to detect exoplanets in the first place. Many surveys rely on the
transit method, by which an exoplanet blocks out part of its host’s light for
some part of its orbit. The probability that a given planet will transit its
host as seen from Earth is generally simply a function of the ratio of the
host’s radius divided by the orbital semi-major axis of the exoplanet,
$R_{*}/a$. However, due to nodal precession, the relative orientation of
exoplanet orbits for a fixed (e.g., Earth-based) observer can shift over time,
allowing previously non-transiting exoplanets to transit at a later date, and
vice-versa. As such, transit probability becomes not just a function of
$R_{*}/a$, but also of the exoplanet’s precession rate and overall time
baseline that the system has been observed. Due to the continuing work of the
various observational surveys, the observation time for significant parts of
the sky has reached well over a decade.

In this work we investigate the time evolution of transit architectures due to
nodal precession cause by rapidly rotating, oblate star, and the transit
probability increase with increasing observation baselines. We provide
equations that describe the time evolution of an exoplanet’s impact parameter,
$b$, and projected obliquity, $\lambda$, and outline which orbital
architectures are most impacted by nodal precession, which ought to improve
future studies of precessing exoplanets. Finally, we estimate how the nodal
precession may impact the statistics of observed exoplanet system
architectures, in particular regarding spin-orbit misalignment.

##  2 Mathematical Methods

Figure 1: Angular momenta of rapidly rotating stars and their planets. The
figure shows the angular momenta of several rapidly rotating stars known to
have short-period planets, compared to their planets’ orbital angular momenta.
The stellar rotational angular momenta, marked as stars, are calculated using
a moment of inertia factor of $\alpha\sim 0.04$, a lower limit consistent with
various models of stellar structure. The orbital angular momenta of the
planets are marked by the plus signs. Model calculations using the stellar
evolution code SSE (Hurley et al., 2000) for the beginning, middle, and end of
the main sequence evolution of stars covering the range of masses used in this
figure are shown as blue, green, and red curves, respectively. The grey curves
show three examples for the orbital angular momentum of a $1$ MJup planet at a
$1.5$, $5$, and $10$ day orbit. Note that Hot Jupiters orbiting stars above
the Kraft break (to the right of the blue bar) generally have much lower
orbital angular momenta than their host stars, as predicted by the model
calculations. An example of a Hot Jupiter orbiting a star below the Kraft
break, 51 Pegasi, for which the planet’s angular momentum dominates, is given
on the left side of the figure. As such, the nodal precession model described
in this work appears generally valid for short-period planets orbiting stars
above the Kraft break. Stellar and planetary parameters used to calculate the
angular momenta are taken from observations (Mayor & Queloz, 1995; Simpson et
al., 2010; Szabó et al., 2012; von Essen et al., 2014; Johnson et al., 2015;
Gaudi et al., 2017; Raynard et al., 2018; Zhou et al., 2019; Wong et al.,
2021).

The impact parameter can be measured from the precise shape of the light curve
alone (e.g., Charbonneau et al. 2000; Seager & Mallén-Ornelas 2003; Carter et
al. 2008), and is related to the inclination angle of the planet’s orbital
angular momentum vector against the line of sight, $i_{p}$, the stellar
equatorial radius $R_{*}$, and orbital semi-major axis $a$ via the equation

$b=(a/R_{*})\cos i_{p}\ .$ (1)

The quantity $a/R_{*}$ can be directly measured from the transit curve and
radial velocity curve (Seager & Mallén-Ornelas, 2003; Winn, 2010; Carter et
al., 2008), and thus it is possible to measure $i_{p}$ purely from
observables. Disregarding the actual size of the transiting object, $b$ must
be between the values of $-1$ and $1$ for transits to occur.

The projected obliquity $\lambda$ is the two-dimensional projection of the
true spin-orbit angle $\psi$. Their relation is described by the equation

$\cos\psi=\cos i_{*}\cos i_{p}+\sin i_{*}\sin i_{p}\cos\lambda\ $ (2)

(e.g., Iorio, 2011), where $i_{*}$ is the stellar spin angle versus the line
of sight, defined such that $i_{*}=0\degree$ indicates that the star is viewed
pole-on (see Fig. 2 for an overview of the transit geometry). The projected
obliquity can be measured a number of says, including the Rossiter-McLaughlin
effect (Rossiter, 1924; McLaughlin, 1924; Queloz et al., 2000; Gaudi & Winn,
2007), Doppler Tomography (Collier Cameron et al., 2010; Johnson et al.,
2017), gravity darkening (Barnes, 2009; Barnes et al., 2011; Ahlers et al.,
2020), and starspots (Désert et al., 2011; Sanchis-Ojeda & Winn, 2011; Dai &
Winn, 2017). See Albrecht et al. (2022) for a thorough review of measurements
of the project obliquity of transiting exoplanets and references therein.

Figure 2: Observable transit geometry as defined in this study. The figure
presents an example orbital geometry of a rotating, oblate star being
transited by a retrograde, inclined planet, shown from two perspectives. The
left image shows the system from the observers perspective, with the star’s
spin axis (solid black line) oriented along the vertical axis. The star’s spin
axis is tilted towards the observer, as highlighted by its equator (dashed
black line). The orbit’s impact parameter $b$ and projected spin-orbit angle
$\lambda$ are shown in red, as well as the orbital path (solid red line,
direction marked with arrows). The right inset image shows an overview of the
observable orbital angles, with the observer’s line of sight being the x-axis
(magenta dotted line, observer to the left, marked by $\bigoplus$). The tilt
of the stellar spin axis (thick black line), i∗, is defined in relation to the
line of sight. The red line shows the planet’s orbital angular momentum
vector, which forms an angle ip with the line of sight. The projection of the
angular momentum vector onto the plane of the sky (the y-z plane), shown by
the green line, forms the angle $\lambda$ with the z-axis, which is the
projection of the stellar spin axis. The true spin-orbit angle $\psi$ is
formed between the stellar spin axis and angular momentum vector and can be
determined via Eq. 2.

These observable quantities are ultimately related to the orbital elements
$\Omega$, the longitude of the ascending node, and $I$, the inclination of the
orbital angular momentum versus the plane of the sky, via the equations

$\tan\Omega=-\sin\lambda\tan i_{p}$ (3)

and

$\cos I=\cos\lambda\sin i_{p}\ ,$ (4)

which allows for an alternate expression for $i_{p}$ in the form of

$\cos i_{p}=\sin I\cos\Omega\ .$ (5)

In this work we are mostly interested in systems where the angular momentum of
the stellar rotation dominates over the orbital angular momentum of its
planet, which is generally the case for short period planets orbiting hot
stars above the Kraft break (Kraft, 1967; Ward et al., 1976), as we show in
Fig. 1. The long-term precession of $\Omega$ and $I$ of any planet with
significantly smaller orbital angular momentum than its host star’s rotational
angular momentum can be described by the equations (e.g., Iorio, 2016)

$\displaystyle\dot{\Omega}=-\frac{3\pi J_{2}R_{*}^{2}}{2Pa^{2}}$
$\displaystyle\\{2\sin{i_{*}}\cos{i_{*}}\cos{2I}\csc{I}\cos{\Omega}$ (6)
$\displaystyle-\cos{I}\left(1-3\sin^{2}{i_{*}}+\cos^{2}{i_{*}}\cos{2\Omega}\right)\\},$

and

$\displaystyle\dot{I}=-\frac{3\pi
J_{2}R_{*}^{2}}{Pa^{2}}\cos{i_{*}}\sin{\Omega}\\{$
$\displaystyle\sin{i_{*}}\cos{I}$ (7)
$\displaystyle-\cos{i_{*}}\sin{I}\cos{\Omega}\\},$

with $P$ being the planet’s orbital period and $J_{2}$ being the star’s
quadrupole gravitational moment.

In the case that the rotating star is observed equator-on $(i_{*}=90\degree)$,
Equations 6 and 7 simplify to the straightforward expressions

$\dot{\Omega}_{eq}=-\frac{3\pi J_{2}R_{*}^{2}}{Pa^{2}}\cos{I_{eq}},$ (8)

and

$\dot{I}_{eq}=0,$ (9)

which are the standard equations for nodal precession in the frame of the
stellar spin, for which $I=I_{eq}=\psi$.

While Equations 6 and 7 are not easily integratable over time, Equations 8 and
9 have straightforward integral solutions in

${\Omega}_{eq}(t)=\Omega_{eq,0}-\left(\frac{3\pi
J_{2}R_{*}^{2}}{Pa^{2}}\cos{I_{eq,0}}\right)t$ (10)

and

${I}_{eq}(t)=I_{eq,0}=\psi={\rm const}.$ (11)

From this it becomes clear that the precession of $\Omega_{eq}$ has a
circulation period of

$t_{P}=\frac{2Pa^{2}}{3J_{2}R_{*}^{2}|\cos{\psi}|},$ (12)

applicable to $\Omega$ and $I$ in all reference frames. Furthermore, we can
construct a new expression for the time evolution of impact parameter $b$ by
combining Equations 5, 10, and 11. As a first step, consider that the maximum
and minimum values of $b$ are reached when $\cos{\Omega}=\pm 1$ and that $I$
can only oscillate between values of $\psi-(90\degree-i_{*})$ and
$\psi+(90\degree-i_{*})$. Since $\Omega$ and $I$ oscillate with the same
period, the extreme values of $b$ are thus given by

$b_{extrema}=\pm\frac{a}{R_{*}}{\sin{(\psi\mp(90\degree-i_{*}))}},$ (13)

which can be simplified to

$b_{extrema}=-\frac{a}{R_{*}}{\cos{(\psi\pm i_{*})}}.$ (14)

The value of $b$ will oscillate between these extreme values with a period of
$t_{P}$, as this is the oscillation period in all reference frames. We thus
construct the equation

$\displaystyle b(t)=\frac{a}{R_{*}}\biggl{\\{}$
$\displaystyle\frac{-\cos{(\psi+i_{*})}-\cos{(\psi-i_{*})}}{2}$ (15)
$\displaystyle+$ $\displaystyle\frac{-\cos{(\psi+i_{*})}+\cos{(\psi-
i_{*})}}{2}$ $\displaystyle\times\cos{\left(\frac{2\pi}{t_{P}}\times
t\right)}\biggr{\\}}.$

Finally, via the trigonometric identities concerning the sum of angles, we
arrive at the expression

$\displaystyle b(t)=\frac{a}{R_{*}}\biggl{\\{}$
$\displaystyle-\cos{\psi}\cos{i_{*}}$ (16)
$\displaystyle+\sin{\psi}\sin{i_{*}}\cos{\left(\frac{2\pi}{t_{P}}\times
t\right)}\biggr{\\}}.$

By numerically integrating Equations 6 and 7, we have verified that this
expression describes the evolution of $b$ for any system and any observing
geometry, assuming one can derive the true spin-orbit angle, $\psi$. However,
the phase of the oscillation has to be adjusted by the correct choice of $t$
depending on observations. The time derivative of $b$ is, consequently,

$\dot{b}(t)=-\frac{a}{R_{*}}\frac{2\pi}{t_{P}}\sin{\psi}\sin{i_{*}}\sin{\left(\frac{2\pi}{t_{P}}\times
t\right)}.$ (17)

For any relevant host star with persistent physical characteristics, the time
evolution of $b$ and $\dot{b}$ thus purely depends on the angle of the stellar
spin axis versus the line of sight, $i_{*}$, and the true spin-orbit angle,
$\psi$. Figure 3 shows several examples of the evolution of $b$ following
Equation 16. For $\lambda$ a similar time evolution expression can be
constructed. First, by rearranging Equation 2, we obtain

$\cos{\lambda}=\frac{\cos{\psi}-\cos{i_{p}}\cos{i_{*}}}{\sin{i_{*}}\sin{i_{p}}}.$
(18)

Since $\cos{i_{p}}=(R_{*}/a)\times b$, we thus arrive at the equation

$\cos{\lambda(t)}=\frac{\cos{\psi}-\frac{R_{*}}{a}b(t)\cos{i_{*}}}{\sin{i_{*}}\sqrt{1-\left[\frac{R_{*}}{a}b(t)\right]^{2}}}.$
(19)

Here, we primarily focus on the time evolution of $b$ as described by Equation
16.

Figure 3: Evolution of the impact parameter $b$ over time. The figure shows
the time evolution of $b$ following Equation 16 for three example geometries
with values of $i_{*}$ and $\psi$ as shown in the figure legend. The three
examples share the same overall system parameters, $a/R_{*}=3.153$,
$J_{2}=3.38\times 10^{-4}$, and $P=127969~{}{\rm s}\simeq 1.48~{}{\rm days}$,
representative of the KELT-9 system (Stephan et al., 2022). The blue curve was
produced using parameters similar to the best-fit solution for the precession
of KELT-9b from Stephan et al. (2022) and is an example of a large amplitude
oscillation of $b$ with clear transit signals for extended times. The cyan
curve shows an example where the planet would oscillate closely around a
grazing transit at $b=1$, while the magenta curve shows an example of a planet
that would never transit. A numerical integration of Equations 6 and 7 yields
identical curves to within a phase offset.

###  2.1 Transit Probability Increase due to Nodal Precession

Figure 4: Fractions of the precession periods that planets will be transiting
their host stars as a function of $i_{*}$ and $\psi$. The figure shows the
likelihood for a planet to transit its host star in the
$\cos{i_{*}}$-$\cos{\psi}$ parameter space, as determined by Equation 16 as a
fraction of its precession period. Only the parameter space inside the black
outline allows transits to occur at some point of the system’s lifetime (see
Equations 21, 22). The figure serves to highlight the general shape of this
function, the exact numerical values are dependent on the stellar and orbital
parameters (see Equations 12 and 16). In this example, $a/R_{*}=3.153$,
$J_{2}=3.38\times 10^{-4}$, and $P=127969~{}{\rm s}\simeq 1.48~{}{\rm days}$,
representative of KELT-9b (Stephan et al., 2022). The average transit fraction
is $0.317=R_{*}/a$.

The expression for $b$ as shown by Equation 16 enables us to recognize certain
geometric relations between $i_{*}$, $\psi$, and the possibility of transits
that make intuitive sense. If the star is viewed pole-on $(i_{*}=0\degree)$,
the orbit’s impact parameter will never change, even though the planet is
precessing. As such, only certain angles of $\psi$ will result in transits,
which consequently will then always be observable. Alternatively, if the star
is viewed equator-on $(i_{*}=90\degree)$, any orientation of $\psi$ will
result in transits for at least some fraction of the precession period, though
precession speed will, of course, still depend on $t_{P}\propto 1/\cos{\psi}$.
As such, if the planet’s orbital plane is aligned with the star’s equator
$(\psi=0\degree)$, while the precession period, $t_{P}$, will be at its
shortest, precession will not be observable, as the amplitude of the change in
$b$ will be zero. If the planet’s orbital plane is exactly aligned with the
stellar rotation axis $(\psi=90\degree)$, the precession period approaches
infinity. Figure 4 shows how these geometric relations translate into transit
fractions over the course of the precession periods. These fractions can be
calculated by determining the times when the impact parameter $b$ crosses the
values of $-1$ or $1$, via the expression

$t(b)=\frac{t_{P}}{2\pi}\arccos{\left(\frac{b\frac{R_{*}}{a}+\cos{\psi}\cos{i_{*}}}{\sin{\psi}\sin{i_{*}}}\right)},$
(20)

and comparing it to the precession timescale. The boundary between the never-
transit and sometimes-transit regions in Figure 4 is formed by geometries
where the minimum value of $b$ during precession is exactly equal $1$ or the
maximum value of $b$ is equal $-1$, such that

$b_{max}=\max\left(-\frac{a}{R_{*}}\cos{(\psi\pm i_{*})}\right)=-1,$ (21)

and

$b_{min}=\min\left(-\frac{a}{R_{*}}\cos{(\psi\pm i_{*}})\right)=1.$ (22)

Figure 5: Increase of the transit likelihood due to extended observation time
baseline. The figure shows how an extended observation period increases the
transit likelihood by allowing significant nodal precession to occur. As such,
the increase is primarily a function of $\cos{\psi}$, as it depends on the
precession period, $t_{P}$. The system parameters are identical to the one in
Fig. 4. The observation baseline used is $15$ years, motivated by the Kepler
and TESS missions. On average, the observation likelihood for a system in this
example increases by about $8\%$, while for the fastest precessing systems
($\psi$ close to $0\degree$ or $180\degree$), the transit likelihood increases
by nearly $19\%$.

While the transit fractions over precession period as shown in Fig. 4 give us
a sense for what values of $\psi$ and $i_{*}$ are more likely to be observed
for any particular system, they do not change the overall transit likelihood
for a random collection of systems over a short observation timescale. This
“instantaneous” transit likelihood still follows $P_{transit}=R_{*}/a$.
However, due to nodal precession, the transit likelihood does indeed change
given a longer observation time period. The Kepler and TESS missions give us
such an extended observation period, for a portion of the sky, on the order of
$15$ years, assuming TESS continues at least until $2024$ CE. As such, the
transit likelihood for any particular system changes to

$P_{transit,new}=\frac{R_{*}}{a}+\frac{t_{obs}}{t_{P}},$ (23)

with $t_{obs}$ being the observation time period. $P_{transit,new}$ naturally
reaches a maximum of $1$ when $t_{obs}/t_{P}$ reaches $1-R_{*}/a$. The transit
likelihood increase is thus also a function of $\psi$, with values of $\psi$
close to $0\degree$ or $180\degree$ resulting in the shortest precession
periods, experiencing the largest increase of their transit likelihood. Fig. 5
shows this increase in transit likelihood in the same phase space as used in
Fig. 4.

We note here that the increase of the observation likelihood is small for most
exoplanets. Only very short-period planets orbiting hot, rapidly rotating
stars (which thus should posses a significant equatorial bulge and large
gravitational quadrupole moment) will experience nodal precession fast enough
to significantly affect the likelihood. For example, for an exoplanet like
WASP-33b (Johnson et al., 2015; Watanabe et al., 2020; Stephan et al., 2022),
assuming an isotropic likelihood distribution for $\cos{\psi}$ and
$\cos{i_{*}}$, a $15$ years observation time period would increase the overall
observation likelihood by $1~{}\%$, and by about $3~{}\%$ for the most
favorable possible orientations. For an exoplanet like KELT-9b (Stephan et
al., 2022), however, the increase is substantially higher over the same time
frame, with an $8~{}\%$ average increase and a $19~{}\%$ increase for the most
favorable orientations (see Fig. 5).

##  3 Discussion

The increase in transit likelihood, as described in the previous section, is
dependent on the observation time frame, spin-orbit angle $\psi$, stellar
$J_{2}$ value, and orbital period of the planet, and creates a set of biases
for transit observations. In general terms, exoplanets orbiting with a spin-
orbit angle $\psi$ close to $0\degree$ or $180\degree$, around stars with
large $J_{2}$ values, on short orbits, are more likely to be observed as
transiting planets at some point during any given observation time baseline,
with the relative proportion of such planets increasing with increasing time
baseline. While these effects ought to be small to negligible for most
exoplanets, for certain classes the effects may be significant, which we
outline here.

###  3.1 Hot Jupiters orbiting Hot Stars

As outlined in Section 2, hot stars with short spin periods and large
gravitational quadrupole moments are the primary environment to cause rapid
nodal precession of close-in planetary orbits. So far, three Hot Jupiters
orbiting such stars have measured precession periods, namely Kepler-13Ab
(about $500$ years, e.g., Szabó et al., 2012), WASP-33b (in the range of $800$
to $1500$ years, e.g., Johnson et al., 2015; Watanabe et al., 2020; Borsa et
al., 2021; Stephan et al., 2022), and KELT-9b (about $900$ years, Stephan et
al., 2022). These planets orbit their host stars with a range of obliquity
values and the stars have $J_{2}$ values in the approximate range of $6\times
10^{-5}$ to $4\times 10^{-4}$. As such, the transit likelihood increases for
these three known precessing Hot Jupiters over the approximately $15$ year
long observation time frame of the Kepler and TESS missions are non-
negligible.

For a planet like Kepler-13Ab, the base transit probability, derived from its
$a/R_{*}$ value of $\sim 4.44$ (e.g., Shporer et al., 2014), is about
$22.5\%$. Given its rapid precession, over a $15$ years observation time frame
this probability increases by $3\%$ to about $25.5\%$. For KELT-9b, the base
transit probability of $31.7\%$ (e.g., Gaudi et al., 2017) increases by
$1.7\%$ to $33.4\%$, and for WASP-33b the base transit probability of $26.4\%$
(e.g., Collier Cameron et al., 2010) is increased by between $1$ and $1.9\%$
up to between $27.4$ and $28.3\%$.

While these increases are overall comparatively small and do not change the
transit likelihoods of these planets in a qualitatively significant way, they
highlight that there potentially exists a systemic underestimation of the
transit likelihoods of Hot Jupiters that may affect population-level studies,
at least around massive, hot, rapidly rotating stars. In particular, applying
the equations outlined in Section 2, one can estimate the potential transition
likelihood increases for the most rapidly precessing orientations of $\psi$.
For the example of a planet like KELT-9b, the potential transit likelihood
increase would be up to about $19\%$, with an average increase for all
possible orientations in the $\cos{\psi}$, $\cos{i_{*}}$ phase space of about
$8\%$ (see Figure 5). Such an increase would be on the order of nearly $1/3$
to $2/3$ of the base transit probability for a planet like KELT-9b,
potentially even on the order of $1$ or larger for a planet like Kepler-13Ab,
significantly impacting observational statistics, assuming that the
distributions of $\cos{\psi}$ and $\cos{i_{*}}$ are truly isotropic. In fact,
as more rapidly precessing planets tend to have orbits more aligned with their
host star’s equator, this effect creates an observational bias against
misaligned, slowly precessing planets. As such, statistical estimates of the
distribution of aligned versus misaligned Hot Jupiter orbits will tend to
overestimate the inherent fraction of aligned systems. This effect will become
increasingly important as the time baseline over which a large fraction of the
sky has been observed increases with various future missions and ground-based
follow-up surveys.

###  3.2 Comparison to Circumbinary Planets

Much of the basic precession physics presented in this work is similar in
nature to the precession observed for circumbinary planets (CBPs) (e.g.,
Martin, 2017). However, the timescales for the precession of close-in orbiting
CBPs is generally significantly shorter than that of short-period planets
around hot stars and is on the order of decades rather than centuries or
millennia, in part due to the much more significant angular momentum of a
tight binary star that is driving the planetary orbit’s nodal precession.
However, the transit geometry is more complicated given by the binary nature
of the to-be-transited host, making analysis of the transit evolution more
complex. Furthermore, the speed of the precession is significantly impacted by
the host stars’ internal structure or tidal mechanics, but mostly by the mass
and orbital configuration of the binary. As such, the two cases complement
each other for these separate planetary populations.

##  4 Summary and Conclusions

In this work we have provided a description of the time evolution of exoplanet
transits due to nodal precession caused by rapidly rotating host stars,
generally valid for short-period planets orbiting stars above the Kraft break.
We derived analytical expressions for the time evolution of the impact
parameter $b$ (see Equations 16 and 17), and estimated the impact of nodal
precession and orbital architectures on increasing transit likelihoods. We can
draw two major conclusions based on our investigation:

  1. 1.

For the most rapidly precessing exoplanets, such as Hot Jupiters orbiting
rapidly rotating massive stars, the observation time frame covered by Kepler
and TESS for parts of the sky has increased the transit likelihoods by a non-
negligible amount on the order of a few to $\sim 10~{}\%$. As such, studies
that estimate planet occurrence rates based on these surveys should take
precession into account for their calculations.

  2. 2.

Due to the dependence of precession rates on the orbital orientation of the
exoplanets, in particular the spin-orbit angle $\psi$ as shown in Equation 12,
planets that are more aligned with their host stars’ spins will be more likely
to be observed over time than misaligned planets. This effect is especially
important for the study of Hot Jupiters, as tidal migration models of their
formation generally predict misaligned or randomly aligned orbits.
Furthermore, different alignment distributions have been observed for low mass
vs. high mass stars, pointing to either different formation mechanisms or
different tidal re-alignment efficiencies between these groups of stars, as
generally expected due to the Kraft break (e.g., Kraft, 1967; Ward et al.,
1976). As such, the dependence of transit likelihood on orbital orientation
may skew conclusions drawn based on observations alone.

Additionally, as significant nodal precession ought to be limited to short-
period planets orbiting hot, young, rapidly-rotating, massive stars,
observations may also eventually overestimate planet occurrence rates for
these types of stars compared to lower-mass and older, slowly rotating ones.

In conclusion, the impact of nodal precession on transit likelihoods will have
to be accounted for when attempting to derive accurate short-period exoplanet
occurrence rates and the distribution of spin-orbit alignment angles. While
the effect is currently relatively small, on the order of a few percent on
average, for certain architectures the effect should already be on the order
of $10$ to $20\%$, thus potentially skewing derived distribution functions.

## Acknowledgments

We would like to thank the anonymous referee for helpful suggestions. We thank
Marshall C. Johnson and David V. Martin for helpful comments and discussions.
A.P.S. acknowledges partial support from the President’s Postdoctoral
Scholarship from the Ohio State University and the Ohio Eminent Scholar
Endowment. A.P.S. and B.S.G. acknowledge partial support by the Thomas
Jefferson Chair Endowment for Discovery and Space Exploration.

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