# Drift Rates of Narrowband Signals in Long-term SETI Observations for
Exoplanets

Jian-Kang Li Department of Astronomy, Yunnan University, Kunming 650500, China
Department of Astronomy, Key Laboratory of Astroparticle Physics of Yunnan
Province, Yunnan University, Kunming 650091, China  Department of Astronomy,
Beijing Normal University, Beijing 100875, China  Hai-Chen Zhao Department of
Astronomy, Beijing Normal University, Beijing 100875, China  Institute for
Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing
102206, China  Zhen-Zhao Tao Department of Astronomy, Beijing Normal
University, Beijing 100875, China  Institute for Frontiers in Astronomy and
Astrophysics, Beijing Normal University, Beijing 102206, China  Tong-Jie
Zhang✉ Department of Astronomy, Beijing Normal University, Beijing 100875,
China  Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal
University, Beijing 102206, China<EMAIL_ADDRESS>Sun Xiao-Hui Department
of Astronomy, Yunnan University, Kunming 650500, China  Department of
Astronomy, Key Laboratory of Astroparticle Physics of Yunnan Province, Yunnan
University, Kunming 650091, China

(Received June 11, 2022; Revised August 19, 2022; Accepted September 07, 2022)

###### Abstract

The Doppler shift of a radio signal is caused by the relative motion between
the transmitter and receiver. The change in frequency of the signal over time
is called drift rate. In the studies of radio SETI (Search for
Extraterrestrial Intelligence), extraterrestrial narrow-band signals are
expected to appear “chirped” since both the exoplanet and the Earth are
moving. Such planet rotation and orbital revolution around the central star
can cause a non-zero drift rate. Other relative motions between the
transmitter and receiver, such as the gravitational redshift and galactic
potential, are negligible. In this paper, we mainly consider the common cases
that the drift rate is contributed by the rotations and orbits of the Earth
and exoplanet in celestial mechanics perspective, and briefly discuss other
cases different from the Earth-exoplanet one. We can obtain the expected
pseudosinusoidal drifting result with long-term observations, shorter orbital
periods of exoplanets. Exoplanets with higher orbital eccentricities can cause
asymmetric drifting. The expected result should be intermittent
pseudosinusoidal curves in long-term observations. The characteristics of
pseudosinusoidal curves, as another new criterion for extraterrestrial
signals, can be applied to long-term SETI reobservations in future research.

Search for extraterrestrial intelligence(2127); Celestial mechanics(211)

††journal: ApJ

##  1 Introduction

Radio frequency waves are ideal electromagnetic waves for interstellar
communication since the radio photons cost less energy and the attenuation
caused by interstellar medium absorption is lower (Cocconi & Morrison 1959).
The Doppler shift of electromagnetic wave is caused by the relative motion of
the transmitter and receiver. In radio SETI researches, we usually apply a
“drift rate” as one of the significant parameters to distinguish
extraterrestrial signals from signals generated on the Earth in radio
observations. For a transmitter emitting signals with constant rest frequency
and located on an exoplanet, the frequency of the received signals should
change over time, mainly due to the rotations and the orbits around the
central stars of the Earth and exoplanet. Such a change over time can be
quantified as a non-zero drift rate.

Most of the previous radio SETI observations and research projects only
applied algorithms or used computing power to search for narrow-band signals
with appropriate drift rates, but with little or no physical consideration.
For example, the observations for various targets in the Breakthrough Listen
(BL) project at different frequency ranges (e.g. Siemion et al. 2013; Price et
al. 2020; Sheikh et al. 2020; Margot et al. 2021; Gajjar et al. 2021; Franz et
al. 2022), defined the maximum values of drift rates, then applied the “tree
deDoppler” algorithm, which was modified from Taylor (1974), to search for
signals with drift rates fitting the characteristics of the exoplanet systems
within the defined range. Other projects like SETI@home (Anderson et al. 2002)
searched for signals drifting within the ranges of all physically possible
situations.

Recently, Sheikh et al. (2019) studied drift rate with astrophysical
considerations. They proposed 9 possible factors that contribute to drift
rate, presented a theoretical discussion on the effect of each factor, and
calculated maximum drift rates for some representative systems. Although the
theoretical discussion was detailed, covering the main physical cases, there
were still some other factors that had not been taken into consideration,
especially the orbit contributions from planets. In this paper, we mainly
discuss the celestial mechanical factors that contribute to the relative
motion of the transmitter and receiver, and derive a more accurate expression
of drift rate for common Earth-exoplanet cases.

In Section 2, we derive the expressions of drift rate over time mainly caused
by planets’ rotational and orbital motion, then illustrate the drift rates of
some exoplanet systems with potential habitable planets. In Section 3, we
discuss some special cases different from common Earth-exoplanet cases.
Section 4 is a discussion on the drift characteristics that maximize the
distinguishability of ETI signals. Finally, a summary and conclusion are given
in Section 5.

##  2 Celestial Mechanics Factors of Drift Rate

###  2.1 Derivation of Drift Rate

The change in frequency and wavelength of light caused by the the relative
motion of the signal source and observer can be explained by the Doppler
effect, which can be expressed by redshift

$z=\frac{\lambda-\lambda_{0}}{\lambda_{0}},$ (1)

where $z$ is the redshift, $\lambda$ represents the wavelength that observer
receives and $\lambda_{0}$ represents the wavelength that the signal source
emits. In general relativity, the Doppler effect should be written into
relativistic form in Minkowski space:

$1+z=\frac{1+v_{s}\cos\alpha/c}{\sqrt{1-v^{2}_{s}/c^{2}}},$ (2)

where $v_{s}$ is the relative velocity in the line-of-sight direction with
respect to the signal source and observer, $c$ is the speed of light and
$\alpha$ is the angle between the line-of-sight direction and the direction of
emission in the observer’s frame. The $\dfrac{1}{\sqrt{1-v^{2}_{s}/c^{2}}}$
term in equation (2) can be expended into Maclaurin series, then equation (2)
can be changed into

$1+z=(1+v_{s}\cos\alpha/c)\sqrt{\sum_{n=0}^{\infty}{(v_{s}/c)^{2n}}}.$ (3)

In nonrelativistic limit (i.e. $v_{s}\ll c$), higher-order terms in Equation
(3) can be omitted, and the approximation can be written as

$z\approx\frac{v_{s}\cos\alpha}{c}.$ (4)

The redshift in wavelength form can be changed into frequency form, then from
Equation (4) we can write

$\frac{1}{1+z}=\frac{\nu}{\nu_{0}}=\frac{c}{c+v_{s}\cos\alpha},$ (5)

where $\nu$ is the frequency that the observer receives and $\nu_{0}$ is the
frequency that the signal source emits. We set the X-axis direction as the
line-of-sight direction. In the case of an extraterrestrial transmitter and a
receiver on the Earth, the orbital radius of an exoplanet is extremely small
compared with the distance between the Earth and exoplanet (the closest
planetary system to the Earth is Proxima Centauri system, which is about 4.22
lt-yr away, while the size of an orbit of exoplanet ranges from $10^{-3}$ to
$10^{3}$ AU111The data of exoplanets can be seen in
https://exoplanetarchive.ipac.caltech.edu/), and we can only receive signals
that cover the area of the Earth’s orbit, which means that the parallax angle
of the Earth’s orbit can be regarded as zero, and consequently the line-of-
sight direction can be regarded to be parallel to the direction of emission,
but in opposite direction (i.e. $\alpha\approx 180^{\circ}$) (See Figure 1).

Figure 1: The coverage of emission from an exoplanet to the Earth (upper) and
the simplified case for extremely small angle between the line of sight
direction and the direction of emission (lower). The area filled with yellow
is the location range where an observer on the Earth can detect signals from
the exoplanet.

And for $v_{s}\ll c$, equation (5) can be simplified by Maclaurin series
omitting higher-order terms:

$\nu=\nu_{0}+\nu_{0}\frac{v_{s}}{c}.$ (6)

Then drift rate can be obtained by deriving equation (6):

$\dot{\nu}=\frac{v_{s}}{c}\frac{d\nu_{0}}{dt},$ (7)

where $\dfrac{dv_{s}}{dt}$ is the total relative acceleration in the line of
sight between the signal source and observer. For simplicity, we mainly
discuss the situation in which the signal is a narrowband simple beacon that
constantly emitted from the transmitter with a constant frequency in the
reference frame of the receiver on Earth. Sheikh et al. (2019) wrote the
relative radial acceleration into a sum of various terms in line-of-sight
direction, $\sum\limits_{i}{\dfrac{dv_{s}}{dt}}$, and pointed out four main
contributing terms: the rotation of the Earth, the orbit of the Earth, the
rotation of the exoplanet where the transmitter is placed, and the orbit of
the exoplanet. The contributions of the oblateness and topology of the Earth
and exoplanet, the galactic potential, the gravitational redshift in the solar
system are negligible. In addition, the following related effects can also be
neglected in this work:

  1. 1.

The rotational period of the Earth tends to be longer when approaching
perihelion, while shorter when approaching aphelion (Meeus 1997). The same
case can be applied to exoplanets with elliptical orbits.

  2. 2.

The tidal interaction with the Moon that causes gradual rotational
deceleration in megayear timescale (Williams 2000). The same case can be
applied to exoplanets if they have moons orbiting around them.

  3. 3.

The axial precession, astronomical nutation and polar motion of the Earth
(Seidelmann & Urban 2010).

  4. 4.

The precession of the perihelion caused by the the curvature of
spacetime(Einstein 1916).

  5. 5.

The orbital motions around the Galactic Center (GC) of the solar system and
extrasolar systems.

###  2.2 Rotation Terms of the Earth and Exoplanets

For a receiver or transmitter on the surface of a rotating planet, applying a
spherical coordinate system, the location of the receiver or transmitter can
be given by radial distance, polar angle, and azimuthal angle written in
$(r,\theta,\varphi)$ 222ISO 80000-2:2019 is used in this paper.
https://www.iso.org/standard/64973.html (See Figure 2).

Figure 2: A certain point in spherical coordinate system, rotating at an
angular velocity $\omega$

The motion of an object on a rotating planet can be simplified as uniform
circular motion, since the oblateness and topology of the planet can be
negligible, as well as the variation of the rotational period (See Figure 3),
and then the velocity can be expressed by

$v_{s,r}=\frac{2\pi r\sin\theta}{P_{r}}\sin\varphi_{r},$ (8)

where $P_{r}$ is the rotation period, $t$ represents time, and
$\varphi=\frac{2\pi t}{P_{r}}$. Then, the acceleration in the line-of-sight
direction can be derived by

$\frac{dv_{s,r}}{dt}=\frac{4\pi^{2}r\sin\theta}{P^{2}_{r}}\cos\varphi_{r}.$
(9)

Figure 3: The motion of an object on a rotating planet and the simplified
case.

####  2.2.1 Rotation of the Earth

In most cases the receivers on the Earth are not located on the equator. The
latitude $\phi$ can be converted to polar angle by $90^{\circ}\pm\phi$, plus
for south semi-sphere and minus for north semi-sphere (See Figure 4). For
example, the latitude of 100m Robert C Byrd Green Bank Telescope (GBT):
$38^{\circ}25^{\prime}59^{\prime\prime}$ N can be converted to
$\theta=51^{\circ}34^{\prime}01^{\prime\prime}$, and the latitude of Parkes
64m Telescope (Parkes): $32^{\circ}59^{\prime}52^{\prime\prime}$ S can be
converted to $\theta=57^{\circ}00^{\prime}08^{\prime\prime}$. Using $\oplus$
as the subscript of the Earth and replacing $r$ with $R$, for a receiver with
polar angle $\theta_{re}$, the acceleration in the line-of-sight direction can
be written as

$\frac{dv_{s,r,\oplus}}{dt}=\frac{4\pi^{2}R_{\oplus}\sin\theta_{\text{re}}}{P_{r,\oplus}^{2}}\cos\varphi_{r,\oplus},$
(10)

where $R_{\oplus}$ is the radius of Earth. Equation (10) is the situation in
which the direction in which transmitter points may not be parallel to the
equator plane. For a target with decl. $\delta$, Equation (10) should be
rewritten as

$\frac{dv_{s,r,\oplus}}{dt}=\frac{4\pi^{2}R_{\oplus}\sin\theta_{\text{re}}\cos\delta}{P_{r,\oplus}^{2}}\cos\varphi_{r,\oplus}.$
(11)

Figure 4: Conversion between latitude and polar angle

####  2.2.2 Rotation of Exoplanets

Similar to Equation (11), the acceleration in line-of-sight direction for a
transmitter on the exoplanet with polar angle $\theta_{tr}$ can be written as

$\frac{dv_{s,r,p}}{dt}=\frac{4\pi^{2}R_{p}\sin\theta_{\text{tr}}\cos\delta^{\prime}_{\oplus}}{P_{r,p}^{2}}\cos\varphi_{r,p},$
(12)

where p is the subscript of exoplanet, $\theta_{tr}$ represents the polar
angle of the transmitter, and $\delta^{\prime}_{\oplus}$ is the decl. of Earth
in the equatorial coordinate system of the exoplanet frame. To maximize the
acceleration in the line-of-sight direction, the transmitter should be set on
the equator of the exoplanet (i.e. $\theta_{\text{tr}}=90^{\circ}$), and the
$\delta^{\prime}_{\oplus}$ should be assumed to be 0.

The rotation period of an exoplanet is extremely tough to determine. Snellen
et al. (2014) measured the rotation period of Beta Pictoris b, the first time
for us to measure the rotation period of an exoplanet. Brogi et al. (2016)
used CO and H2O spectra to measure the rotational velocity of HD 189733 b.
However, the methods used in the above studies can be only applied to the
cases of hot Jupiter, instead of Earth-like exoplanet. Another method to
determine the rotation period of an exoplanet is tidal locking or spin–orbit
resonance. Moreover, Barnes (2017) implied that tidally locked exoplanets are
common in Milky Way and habitable exoplanets are possible to be tidally locked
by their host stars. TRAPPIST-1 system is a perfect example since all the
seven planets are likely to have been tidally locked by TRAPPIST-1 (Gillon et
al. 2017), and some of them are located in the habitable zone (O’Malley-James
& Kaltenegger 2017; Airapetian et al. 2020). Makarov et al. (2012) also
pointed out that Gliese 581 d, a potentially habitable exoplanet, is extremely
probable to be trapped into spin–orbit resonance.

###  2.3 Orbit Terms of the Earth and Exoplanets

The orbits of most planets are elliptical rather than perfectly circular.
Applying a polar coordinate system, the orbit of a planet around its host star
can be described by radial distance and polar angle written into ($r,\varphi$)
(See Figure 5). Since the mass of a planet is several orders of magnitude
smaller than its host star, the orbit of a planet can be considered as an
ellipse around its host star.

Figure 5: The orbit of a planet around its host star in polar coordinate
system.

In a polar coordinate system, the radial distance can be written as

$r=\frac{a\left(1-e^{2}\right)}{1+e\cos\varphi_{o}},$ (13)

where $e$ is the eccentricity of the orbit and $a$ is the length of the
semimajor axis. The velocity vector and acceleration vector can be given by

$\vec{v}=\dot{r}\hat{e}_{r}+r\dot{\varphi}_{o}\hat{e}_{\varphi},$ (14)
$\vec{a}=\left(\ddot{r}-r\dot{\varphi}_{o}^{2}\right)\hat{e}_{r}+\left(r\ddot{\varphi}_{o}+2\dot{r}\dot{\varphi}_{o}\right)\hat{e}_{\varphi},$
(15)

where $\hat{e}_{r}$ is the radial unit vector and $\hat{e}_{\varphi}$ is the
angular unit vector. $v_{r}=\dot{r}$ and $a_{r}=\ddot{r}-r\dot{\varphi}^{2}$
are the radial velocity and radial acceleration,
$v_{\varphi}=r\dot{\varphi}_{o}$ and
$a_{\varphi}=r\ddot{\varphi}+2\dot{r}\dot{\varphi}$ are the circumferential
velocity and circumferential acceleration, respectively. For the case of a
central force field (gravitational field), only the radial acceleration
exists, written as

$\displaystyle a_{r}$
$\displaystyle=\ddot{r}-r\dot{\varphi}_{o}^{2}=\frac{GM_{\text{star}}}{r^{2}},$
(16a) $\displaystyle a_{\varphi}$
$\displaystyle=r\ddot{\varphi}_{o}+2\dot{r}\dot{\varphi}_{o}=0,$ (16b)

where $G$ is the gravitational constant and $M_{\text{star}}$ is the mass of
central star. Then, the velocity and acceleration in the line-of-sight
direction can be written by

$v_{s,o}=\dot{r}\cos\varphi_{o}-r\dot{\varphi}_{o}\sin\varphi_{o},$ (17)
$\displaystyle\frac{dv_{s,o}}{dt}$
$\displaystyle=\left(\ddot{r}-r\dot{\varphi}_{o}^{2}\right)\cos\varphi_{o}-(r\ddot{\varphi}_{o}+2\dot{r}\dot{\varphi}_{o})\sin\varphi_{\text{o}}$
(18) $\displaystyle=\frac{GM_{\text{star}}}{r^{2}}\cos\varphi_{o}.$

Equation (13) does not give the relation between polar coordinate and time,
since the angular velocity of ellipses’ orbital motion is not uniform. Kepler
(1609) gave an alternative form, written as

$\left\\{\begin{array}[]{l}r=a\left(1-e\cos E\right),\\\
\varphi_{o}=2\arctan\left(\sqrt{\frac{1-e}{1+e}}\tan\frac{E}{2}\right),\\\
\end{array}\right.$ (19)

where $E$ is the eccentric anomaly (See Figure 6).

Figure 6: Eccentric anomaly is one of the angular parameters defined by a
planet orbiting a central star along an elliptic Kepler orbit.

The conversion between $E$ and $\varphi$ can be given by

$\cos\varphi_{o}=\frac{e-\cos E}{e\cos E-1},\quad\sin
E=\frac{\sqrt{1-e^{2}}\sin\varphi_{o}}{1+e\cos\varphi_{o}}.$ (20)

The Kepler equation (Kepler 1609) gives the relation between $E$ and $t$ by
mean anomaly $M=2\pi t/P$, which can be written as

$M=\frac{2\pi t}{P_{o}}=E-e\sin E.$ (21)

Colwell (1992) gave a Bessel’s solution of Kepler equation by extending
Equation (21) into Fourier sine series:

$E=\frac{2\pi
t}{P_{\text{o}}}+\sum_{n=1}^{\infty}{\frac{2}{n}J_{n}\left(ne\right)\sin\left(\frac{2n\pi
t}{P_{\text{o}}}\right)},$ (22)

where $J_{n}\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}{\cos\left(nE-x\sin
E\right)dE}$ is the Bessel integral in modern notation. Then, the radial
velocity and circumferential velocity can be derived by Equations (19), (21)
and (22):

$\displaystyle\dot{r}=\frac{2\pi
a}{P_{o}}\left[\sum_{n=1}^{\infty}{\frac{2}{n}J_{n}\left(ne\right)\sin\left(\frac{2n\pi
t}{P_{o}}\right)}\right]\left[1+2\sum_{n=1}^{\infty}{J_{n}\left(ne\right)\cos\left(\frac{2n\pi
t}{P_{o}}\right)}\right],$ (23a) $\displaystyle r\dot{\varphi}_{o}=\frac{2\pi
a\sqrt{1-e^{2}}}{P_{o}}\left[1+2\sum_{n=1}^{\infty}{J_{n}}\left(ne\right)\cos\left(\frac{2n\pi
t}{P_{o}}\right)\right].$ (23b)

####  2.3.1 Orbit of the Earth

For a receiver located on Earth, the velocity is contributed by rotational
velocity $v_{\text{r},\oplus}$ and orbital velocity $v_{\text{o},\oplus}$.
However, the direction that points at the target is probably not parallel to
the ecliptic plane. Therefore, the contribution from orbital velocity and
should be its projection on equator. Rewriting Equations(17), (18) and (19) by
using $\oplus$ as the subscript of Earth, the polar coordinate of Earth,
velocity and acceleration in line-of-sight direction can be written as

$\left\\{\begin{array}[]{l}r_{\oplus}=\frac{a_{\oplus}\left(1-e_{\oplus}^{2}\right)}{1+e_{\oplus}\cos\varphi_{o,\oplus}}=a_{\oplus}\left(1-e_{\oplus}\cos
E_{\oplus}\right),\\\
\varphi_{o,\oplus}=2\arctan\left(\sqrt{\frac{1-e_{\oplus}}{1+e_{\oplus}}}\tan\frac{E_{\oplus}}{2}\right),\\\
\end{array}\right.$ (24)
$v_{s,o,\oplus}=(\dot{r}_{\oplus}\cos\varphi_{o,\oplus}-r_{\oplus}\dot{\varphi}_{o,\oplus}\sin\varphi_{o,\oplus})\cos\beta,$
(25)
$\frac{dv_{s,o,\oplus}}{dt}=\frac{GM_{\odot}}{r^{2}_{\oplus}}\cos\varphi_{o,\oplus}\cos\beta.$
(26)

where $\beta$ is the ecliptic latitude of the target. The ecliptic coordinate
of exoplanet is usually not provided; the NASA/IPAC Extragalactic Database
provides a tool for the conversion of different astronomical coordinate
systems 333https://ned.ipac.caltech.edu/coordinate_calculator. The input data
are the right R.A. and decl., and the output is the ecliptic longitude and
ecliptic latitude.

####  2.3.2 Orbit of Exoplanets

Since we have little or even no information about the axial tilt of
exoplanets, here the angle between the rotational axis and orbital axis is set
to be 0. For observers on the Earth, there may be an angle between the orbital
plane and the plane perpendicular to the line of sight (i.e. the inclination
of the exoplanet). Similar to Equations (24), (25) and (26), the polar
coordinate of the exoplanet, velocity and acceleration in the line-of-sight
direction can be written as

$\left\\{\begin{array}[]{l}r_{p}=\frac{a_{p}\left(1-e_{p}^{2}\right)}{1+e_{p}\cos\varphi_{o},p}=a_{p}\left(1-e_{p}\cos
E_{p}\right),\\\
\varphi_{o},p=2\arctan\left(\sqrt{\frac{1-e_{p}}{1+e_{p}}}\tan\frac{E_{p}}{2}\right),\\\
\end{array}\right.$ (27)
$v_{s,o,p}=(\dot{r}_{p}\cos\varphi_{o,p}-r_{\text{p}}\dot{\varphi}_{o,p}\sin\varphi_{o,p})\sin
i,$ (28)
$\frac{dv_{s,o,p}}{dt}=(\frac{GM_{star}}{r^{2}_{\text{p}}}\cos\varphi_{o,p})\sin
i,$ (29)

where $M_{star}$ is the mass of the central star and $i$ is the inclination of
the exoplanet. Fischer et al. (2014) suggested the probability of a random
orbital inclination between 30∘ and 90∘ is roughly 87%.

###  2.4 Contributions of the Four Terms

Combining Equations (7), (10), (12), (26) and (29) together, the drift rate
caused by the rotations and orbits of the Earth and exoplanet can be expressed
by

$\dot{\nu}=\frac{\nu_{0}}{c}\left(\frac{dv_{\text{s,r,}\oplus}}{dt}+\frac{dv_{\text{s,r,p}}}{dt}+\frac{dv_{\text{s,o,}\oplus}}{dt}+\frac{dv_{\text{s,o,p}}}{dt}\right),$
(30)

where $\delta$ is the decl. of the exoplanet system. For simplicity, the
initial $\varphi_{r}$ and $\varphi_{o}$ are assumed to start from zero. For an
orbit with a relatively small eccentricity, the orbital motion can be
approximated as uniform circular motion, written as

$\frac{dv_{s,o}}{dt}=\frac{GM_{\text{star}}}{r^{2}}\cos\varphi_{o}\sin
i,\quad\varphi_{o}=\frac{2\pi t}{P_{o}}.$ (31)

It should be noticed that Equation (31) is applied to the signals observed by
Earth that are within Milky Way. For extragalactic signals, other factors such
as the redshift/blueshift and orbits around the galactic centers should be
considered, making Equation (31) more complex.

###  2.5 Cases of Habitable Exoplanets

The Five-hundred-meter Aperture Spherical radio Telescope (FAST) is the
largest single aperture radio telescope in the world (Li & Pan 2016), covering
57% of the celestial sphere. As the most sensitive radio telescope in the
world (Jiang et al. 2020;Wang et al. 2021a), FAST has carried out its first
SETI observation by drift-scan survey (Zhang et al. 2020) and the first
targeted SETI observation (Tao et al. 2022). Since FAST has its unique
advantages of large sky coverage and high sensitivity, we use FAST as an
example of receivers on the Earth in the following calculations. Its latitude
is $25^{\circ}39^{\prime}10^{\prime\prime}$ N
($\theta=64^{\circ}20^{\prime}50^{\prime\prime}$) and its operating frequency
bands covers the range of 70 MHz – 3 GHz (Li & Pan 2016). For transmitters on
the equators of exoplanets, we take three known exoplanets: TRAPPIST-1 g,
Kepler-438 b and Proxima Centauri b, which are potential habitable planets, as
examples. The rotational and orbital parameters of Earth are listed in Table
1, and the rotational and orbital parameters of three exoplanets are listed in
Table 2. The axial tilts of three exoplanets are assumed to be 0 for
maximizing the drift rate.

Table 1: The Rotational and Orbital Parameters of the Earth.

Parameters | Earth
---|---
$R$ |  $6.3781\times 10^{6}$m
$P_{\text{r}}$ | 86164.09890369732 s
$e$ | 0.0167086
$a$ |  $1.4958023\times 10^{11}$ m
$P_{\text{o}}$ |  $3.15581\times 10^{7}$ s
$\varepsilon$ | $23.43641^{\circ}$
$M_{\oplus}$ |  $1.989\times 10^{30}$ kg

Note. The parameters of the Earth and Sun are from Mamajek et al. (2015,
radius), Aoki et al. (1982, rotational period), Simon et al. (1994,
eccentricity, semimajor axis and orbital period), Abalakin (1981, axial tilt),
Cox (2000, mass of Sun) .

####  2.5.1 TRAPPIST-1 g

TRAPPIST-1 g is one of the planets located inside the habitable zone of
TRAPPIST-1 system (O’Malley-James & Kaltenegger 2017). It was discovered by
the Spitzer Space Telescope in 2017 (Gillon et al. 2017). Its lower
eccentricity also indicates that the variation of its distance to the host
star is small, giving a relatively stable climate to TRAPPIST-1 g. Since the
size of TRAPPIST-1 planetary system is significantly small and all the planets
are close enough to TRAPPIST-1, it is believed that all the planets are
tidally locked by the host star (e.g. Gillon et al. 2017; Dong et al. 2018).
Under such an assumption, the rotational period of TRAPPIST-1 g should be
equal to its orbital period. For the transmitter operating on the surface of
TRAPPIST-1 g at 1420MHz, the maximum drift rate we detected is 1.3294 Hz/s.

####  2.5.2 Kepler-438 b

Kepler-438 b was discovered by the Kepler Space Telescope in 2015. Although
there are still debates on whether it is habitable because of the periodic
radiation from its host star (Armstrong et al. 2016), it has still been
studied and discussed in many researches of planetary habitability for its
location inside the habitable zone of Kepler-438 and its high similarity to
the Earth (Kashyap Jagadeesh et al. 2017). Similar to most habitable planets
orbiting around M dwarfs at close distances, Kepler-438 b is also believed to
be tidally locked by its host star (Barnes 2017). Under such an assumption,
the rotational period of Kepler-438 b should be equal to its orbital period.
For the transmitter operating on the surface of Kepler-438 b at 1420 MHz, the
maximum drift rate we detect is 0.709 Hz/s.

####  2.5.3 Porxima Centauri b

Proxima Centauri b is one of the closest exoplanets to our Solar system we
have ever known, discovered by Anglada-Escudé et al. (2016). It is also
located inside the habitable zone of Porxima Centauri. Ribas et al. (2016)
suggested that Porxima Centauri b is not likely to be tidally locked. Its
rotation becomes stable in a 3:2 spin-orbit resonance for an eccentricity
greater than 0.06–0.07, and a 2:1 spin-orbit resonance for the case that
$e=0.16$. In this work, a 2:1 spin-orbit resonance is applied, for higher
variation of the orbital radius. Since Proxima Centauri b was detected by
Doppler spectroscopy, there is still no data for the inclination of Proxima
Centauri b. In this work, the inclination is assumed as 90∘ for maximizing the
drift rate. For the transmitter operating on the surface of Proxima Centauri b
at 1420 MHz, the maximum drift rate we detect is 2.1536 Hz/s.

Table 2: The Rotational and Orbital Parameters of TRAPPIST-1 g, Kepler-438 b
and Porxima Centauri b.

Parameters | TRAPPIST-1 g | Kepler-438 b | Porxima Centauri b
---|---|---|---
$R$ | $1.148_{-0.033}^{+0.032}R_{\oplus}$ | $1.12\pm 0.16R_{\oplus}$ | $1.30_{-0.62}^{+1.20}R_{\oplus}$
$P_{\text{r}}$aaThe rotational periods of TRAPPIST-1 g and Kepler-438 b are considered as the same as their orbital periods under the assumption of tidal locking, and the rotational period of Porxima Centauri b should be half of its orbital period for spin-orbital resonance of 2:1 (Ribas et al. 2016). |  $12.352446$ days | 35.23319 days | 5.5593 days
$e$ | $0.00208\pm 0.00058$ | $0.03_{-0.03}^{+0.01}$ | 0.16
$a$ | $0.04687692\pm 3.2\times 10^{-7}\text{AU}$ |  $0.166^{+0.051}_{-0.042}$AU |  $0.04856\pm 0.00030$AU
$P_{\text{o}}$ |  $12.352446\pm 0.000054$ days |  $35.23319_{-0.00025}^{+0.00029}$ days |  $11.1868_{-0.0031}^{+0.0029}$ days
$M_{\text{star}}$ | $0.0898\pm 0.0023M_{\odot}$ | $0.544^{+0.041}_{-0.061}M_{\odot}$ | $0.1221\pm 0.0022M_{\odot}$
$i$ | $89.721_{-0.026}^{+0.019}(^{\circ})$ | $89.86^{+0.14}_{-0.32}(^{\circ})$ |  $90^{\circ}$bbThe inclination of Porxima Centauri b is assumed to be 90∘ for maximizing the drift rate.
$\alpha$ | $23^{h}06^{m}29.283^{s}$ | $18^{h}46^{m}34.9970^{s}$ | $14^{h}29^{m}42.94853^{s}$
$\delta$ | $-05^{\circ}02^{\prime}28.59^{\prime\prime}$ | $+41^{\circ}57^{\prime}03.9233^{\prime\prime}$ | $-62^{\circ}40^{\prime}46.1631^{\prime\prime}$
$\beta$ccThe ecliptic latitudes of the exoplanet systems is converted by the Coordinate Calculator of NASA/IPAC Extragalactic Database. | $0^{\circ}38^{\prime}0.82^{\prime\prime}$ | $64^{\circ}33^{\prime}55.48^{\prime\prime}$ | $-44^{\circ}45^{\prime}47.87^{\prime\prime}$

Note. The parameters of TRAPPIST-1 and TRAPPIST-1 g are from Grimm et al.
(2018, radius, eccentricity and semimajor axis), Agol et al. (2021, orbital
period and mass of TRAPPIST-1), Delrez et al. (2018, inclination), Cutri et
al. (2003, R.A. and decl. of the system). The parameters of Kepler-438 and
Kepler-438 b are from Torres et al. (2015, radius, eccentricity, semimajor
axis, orbital period, mass of Kepler-438 and inclination), Gaia Collaboration
et al. (2018, R.A. and decl. of the system). The parameters of Porxima
Centauri and Porxima Centauri b are from Tasker et al. (2020, radius), Ribas
et al. (2016, eccentricity), Faria et al. (2022, orbital period and semimajor
axis), Kervella et al. (2017, mass of Porxima Centauri), van Leeuwen (2007,
R.A. and decl. of the system).

####  2.5.4 Calculation Results

There is a special waveband extending from 1420 to 1720 MHz that embraces the
hydrogen and hydroxyl spectral lines in SETI research that is called the
“water hole” (Ross 2009). Table 3 lists the calculation results for the
maximum drift rate created by Earth and the above three exoplanets at the
frequency of 1420 MHz.

Table 3: Calculation Results for the Maximum Drift Rate Created by Earth and
the Above Three Exoplanets at the Frequency of 1420 MHz.

Exoplanet | Rotation terms | Orbit terms | $\dot{\nu}$
---|---|---|---
TRAPPIST-1 g | 0.1454 | 1.1804 | 1.3258
Kepler-438 b | 0.1078 | 0.6012 | 0.7090
Porxima Centauri b | 0.0732 | 2.0804 | 2.1536

Note. Rotation terms are the drift rate created by the rotation of Earth and
exoplanets. Orbit terms are the drift rate created by the orbit of Earth and
exoplanets.

Figure 7 illustrates the frequency change from 1420MHz of the above discussed
exoplanets. At the beginning of the simulated long-term observation, all terms
of accelerations are at maximum while the velocity terms are at minimum. The
slope of the curve is the corresponding drift rate at a certain point.

Figure 7: Pseudosinusoidal relative frequency curves of long-term observations
for TRAPPIST-1 g, Kepler-438 b and Proxima Centauri b.

##  3 Some Different Cases

###  3.1 Interferometers

The above discussions are easily applied to the observations of single dish.
For small telescope arrays, Equation (31) can be also applied to their
observations since the latitudes of each telescope are the same. For large
telescope arrays or interferometers with long baseline, such as Square
Kilometer Array (SKA) and Event Horizon Telescope (EHT), equation (31) can not
be applied to the observations, since the locations of the telescopes (arrays)
are significantly different. And the Earth’s rotation term
$\frac{dv_{s,r,\oplus}}{dt}$ should be correspondingly changed.

For large telescope arrays or interferometers with long baseline, Equation
(11) should be rewritten as

$\frac{dv_{{s,r,}\oplus}}{dt}=\frac{4\pi^{2}R_{\oplus}\sin\theta_{\text{re},i}\cos\delta}{P_{{r},\oplus}^{2}}\cos\varphi_{{r},\oplus},$
(32)

where $\theta_{\text{re},i}$ is the polar angle of the $i$th telescope. If a
signal is actually a extraterrestrial signal, when a pair of telescopes
receive this signal, as Figure 8 illustrates, the wave of the signal should
travel an extra distance $\boldsymbol{b}\cdot\hat{s}=b\cos\theta_{ij}$, and
the output frequency curves of the two telescopes should have a geometric
delay

$\tau_{g,ij}=\frac{b\cos\theta_{ij}}{c},$ (33)

where $\boldsymbol{b}$ is the baseline vector with length $b$, $\hat{s}$ is
the unit vector in the direction that the telescopes point at, and
$\theta_{ij}$ is the angle between $\boldsymbol{b}$ and $\hat{s}$. Such
geometric delay should cause a phase difference between two curves
$\Delta\phi_{ij}$:

$\Delta\phi_{ij}=2\pi\nu_{0}\tau_{g,ij}.$ (34)

The amplitudes of the curves and the maximum drift rate may have small
difference between each other since the telescopes may be located at different
latitudes (i.e. the polar angle $\theta_{\text{re},i}$ and
$\theta_{\text{re},j}$ may not be equal). The criteria of phase and amplitude
differences can be used to determine whether a signal is an ETI signal.

Figure 8: Left: extraterrestrial signal received by a pair of telescopes
pointing at the same direction $\hat{s}$. The baseline vector $\boldsymbol{b}$
points from the $j$th telescope to $i$th telescope. Right: frequency drifts of
the signal received by the two telescopes. The different locations of two
telescopes cause the geometric delay of the signal.

###  3.2 Spacecrafts, Space Probes and Interstellar Probes

Transmitters that emit ETI signals may not be located on the surface of an
exoplanet. A transmitter may be launched as a satellite orbiting an exoplanet
or a host star and emits signals in all directions. For the case that a
satellite orbits an exoplanet, its velocity and acceleration can be written as

$v_{s,t}=\sqrt{\frac{GM_{p}}{\left(R+h\right)}}\sin\varphi_{r},$ (35)
$a_{s,t}=\frac{GM_{p}}{\left(R+h\right)^{2}}\cos\varphi_{r},$ (36)

where $M_{p}$ is the mass of the exoplanet, $h$ is the height of the
satellite. The orbital period can be calculated by the Kepler’s third law:

$\frac{\left(R+h\right)^{3}}{P^{2}}=\frac{GM_{p}}{4\pi^{2}}.$ (37)

For the case in which a satellite orbits a host star, the rotation term can be
omitted. Similar to the orbit terms of the Earth and exoplanets, the
nonuniform motion of an elliptic orbit should also be taken into
consideration.

A space probe launched into the space is able to escape its planetary system,
such as Voyager 1. Interstellar probes have already escaped their planetary
systems and are travelling in the interstellar space. For these two cases,
both the rotation terms and orbit terms can be omitted. What we need to
consider is the velocity in the line of sight. The velocity of an interstellar
probe is significantly greater than space probes. Lubin (2016) proposed an
envisioned interstellar probe with a laser phased array as energy propulsion.
The maximum velocity can reach more than $c$/4 in a few minutes. And the
acceleration should gradually decline from $\sim 10^{5}$ m/s2 to $\sim
10^{-11}$ m/s2 after reaching the maximum velocity. Such a decrease of
acceleration can cause a remarkable drift rate, which gradually decreases to a
negligible level over time. In comparison, Voyager 1 spacecraft was launched
in 1977, its velocity has been increased to 17 km/s after 37 yr of flight. For
a spacecraft with a slow and uniform velocity like Voyager 1, its motion can
cause a Doppler shift of an emitted signal, but with zero drift rate, since
the acceleration is zero for uniform motion.

###  3.3 Signal from the Galactic Center

The GC is considered to contain a large number of habitable planets owing to
the higher number density of stars. The model proposed by Gowanlock et al.
(2011) and the following work by Morrison & Gowanlock (2015) and Gajjar et al.
(2021) inferred that the GC is an ideal region for emergence of life and for
technosignature detection. Carroll-Nellenback et al. (2019) suggested that
higher densities of stars and habitable planets are beneficial to the
development of advanced space-faring societies. Simulations of Davari et al.
(2022) also revealed that about half of planets orbiting around the
Sagittarius A* cluster (S stars), a cluster of stars in close orbit around
Sagittarius A* (Sgr A*), can survive in the unstable environment in the GC.

Assuming that space-faring societies have formed in the GC. They would
communicate and travel between planets orbiting S stars. The motion of S stars
can be regarded as elliptical orbital motion. Until now, S4716 is the closest
star orbiting Sgr A* that we found, with estimated semimajor axis lenght of
$1.93\pm 0.02\times 10^{-3}$ pc, eccentricity of $0.756\pm 0.02$ and the
orbital period of 4 years (Peißker et al. 2022). The velocity and acceleration
of S4716 can be calculated by Equation (17) and (18), and the maximum
acceleration of S4716 should be 2.5753 m/s2. However, we have no information
whether there are planets actually orbiting the S-stars and their orbital
parameters. For simplicity, we assume that the orbital and rotational
contributions of the exoplanets orbiting S-star can be neglected. For our
solar system, the velocity orbiting GC is $\left(233.6\pm 2.8\right)\times
10^{3}$ m/s at a distance to GC $R_{0}=8.112\pm 0.031$ kpc (Mróz et al. 2019),
resulting in an orbital period of $P=2.13754\times 10^{8}$ years. And the
acceleration toward GC is $\left(2.32\pm 0.16\right)\times 10^{-10}$ m/s2
(Gaia Collaboration et al. 2021). Hence the contribution from the orbit of
solar system can be neglected even in galactic scale. For a transmitter
operating on the planet orbiting the S4716, if the the orbital and rotational
contribution of the planet can be neglected, the the maximum drift rate caused
by the orbit around Sgr A* of S4716 is 12.1898 Hz/s. Since the orbits of the
stars of Sgr A* cluster are different from each other, the drift rates of the
signals we receive should be different. High drift rate and the drift rate
difference of the signal can be the criteria of ETI signals from the GC.

The complex environment of the GC also provide the opportunity to detect other
natural astrophysical phenomena, such as pulsars, magnetars, radio transients
or other mysterious phenomena. The pulsars in the GC can give us access to
study the magnetoionic material, the gravitational potential, and star
formation in the GC (Cordes & Lazio 1997). The study of magnetars in the GC
can provide us some key information about the origin of fast radio bursts.
Wang et al. (2021b) found a highly polarized radio source ASKAP
J173608.2-321635 in the GC, and our current theories can not fully explain the
observational results for this source. The BL observation of Gajjar et al.
(2021) also included these natural astrophysical phenomena as one of the
science goals.

###  3.4 Radio Frequency Interference

A signal can be defined as narrowband signal if its bandwidth $B$ are
significantly smaller than the central frequency $\nu_{0}$ ($B/\nu_{0}\ll 1$),
otherwise it should be defined as a broadband signal. Radio frequency
interference (RFI) consists of artificial signals emitted from electronic
devices. Narrowband RFI is the most common kind of RFI, which can be divided
into zone RFI and drifting RFI. Zone RFI maintains stable frequency, mainly
caused by televisions, radio broadcasts, cell phones and satellites, while the
frequency of drifting RFI varies over time, mainly caused by mobile devices
((Zhang et al., 2020)). Narrowband RFI can be expressed as the sum of
sinusoids (Stankovic et al. 2013), written into

$I_{NB}\left(t,\tau\right)=\sum_{n=1}^{N}{a_{n}\left(t,\tau\right)}e^{i\left(2\pi
f_{n}t+\phi_{n}\right)},$ (38)

where $a_{n}\left(t,\tau\right)$, $f_{n}$ and $\phi_{n}$ are the complex
envelope, frequency, and initial phase of the n-th interference, $t$ and
$\tau$ represent fast time and short time respectively. Broadband RFI is not
frequently seen in observations, but it can contaminate the frequency bins and
signals if it appears in the observational data. There are two types of
broadband RFI: chirp-modulated (CM) broadband RFI and sinusoidal-modulated
(SM) broadband RFI (Fan et al. 2019), CM broadband RFI can be expressed as:

$I_{\text{CM}}\left(t,\tau\right)=\sum_{n=1}^{N}{a_{n}\left(t,\tau\right)e^{i\left(2\pi
f_{n}t+\pi\gamma_{n}t^{2}\right)}},$ (39)

where $a_{n}\left(t,\tau\right)$, $f_{n}$, $\gamma_{n}$ indicate the complex
envelope, frequency, and the chirp rate of the n-th interference, while SM
broad-band RFI can be expressed as:

$I_{\text{SM}}\left(t,\tau\right)=\sum_{n=1}^{N}{a_{n}\left(t,\tau\right)e^{i\beta_{n}\sin\left(2\pi
f_{n}t+\phi_{n}\right)}},$ (40)

where $a_{n}\left(t,\tau\right)$, $\beta_{n}$, $f_{n}$, $\phi_{n}$ indicate
the complex envelope, modulation factor, frequency, and the initial phase of
the n-th interference, respectively.

Such sums of complex sinusoids should be similar to the results of
extraterrestrial intelligence (ETI) signals, making it difficult for us to
separate ETI signals and RFI. For RFI consisting of stationary $I_{\text{st}}$
and drifting part $I_{\text{ns}}$: $I(t)=I_{\text{st}}+I_{\text{dr}}(t)$, the
received signal may appear a pseudosinusoidal curve if the drifting RFI
follows Equation (38), Equation (39), or Equation(40). An exception is the
zone RFI created by stationary sources on the Earth, since there is no radial
acceleration causing drift, resulting in a zero drift rate (See Figure 9). The
result in Figure 9 is a hypothetical situation that RFI is uninterrupted for
about 35 days. Actually, the duration of RFI is probably within microseconds
to hours.

Figure 9: Summed pseudosinusoidal relative frequency curves of long-term
observations for uninterrupted random-generated zone RFI, narrowband RFI, CM
broadband RFI and SM broadband RFI.

Some observational or computational methods have been applied to RFI removal.
A common observational method in SETI research is the “On-Off" observational
strategy, which was proposed in the BL project by Isaacson et al. (2017), and
has been applied to the following observations and researches in the project.
Each observation for primary target is followed by an observation for off-
source target which is different from the primary target. The sample of off-
source target stars is from the Hipparcos catalog. ETI signals are expected to
only appear in “On" observations while RFI should appear in both “On" and
“Off" observations. Such observational strategy is often applied to single
dish observations since it is relatively easy for single dish to switch
observation targets, but difficult for interferometric arrays. Other specific
methods such as cross correlation (Abdelgelil & Minn 2019) and spatial
filtering (van der Veen & Boonstra 2016) are uesd by interferometric arrays
use to mitigate RFI. Nebula is a back-end data analysis system originally
developed and used in SETI@home
444https://setiathome.berkeley.edu/nebula/index.php for RFI removal. It was
also applied to the first SETI observations of FAST (Zhang et al. 2020). The
hits in different frequency bins is marked and removed after the full set of
raw data goes through the identification methods of zone FRI, drifting RFI and
multibeam RFI in the process of Nebula RFI removal. More than 99% RFI are
removed by Nebula, and the rest are removed by k-Nearest-Neighbor (KNN)
algorithm.

##  4 Discussion

The forms of received signals, if they are actually from ETI, should appear to
be pseudosinusoidal curves after removing RFI from the data, due to the
rotational and nonuniform orbital motions of the Earth and exoplanets.
Therefore, we can distinguish ETI signals by the characteristics of
pseudosinusoidal curves. Since drift rates are caused by the periodic motions
of planets, in order to search for such signals in our observations, we need
to focus on the observational periods and the periods of the exoplanets,
especially the orbital ones.

For shorter observation periods (several minutes), the frequency variation,
for most of time, can be approximately regarded as a linear function expressed
as $\nu=\nu_{0}+\dot{\nu}t$, where the drift rate can be regarded as a
constant (See Figure 10). Since the observation periods are significantly
shorter than the orbital periods of exoplanets, the possibility that the
observation periods coincide with the phases of the peaks or troughs is less
than other phases. Particularly, in spite of the stationary frequencies of
signals from satellites, the received signals may still vary overtime, as the
satellites may slowly pass through the observation area of the telescope,
leaving linearly varying signals in the observational results similar to
Figure 10.

Figure 10: A simulated short-term observation for a radio source with
$\dot{\nu}=-0.05$ Hz/s.

Any drift rate smaller than the minimum drift rate $\dot{\nu}_{\min}$ is
indistinguishable from the zero drift rate. $\dot{\nu}_{\min}$ can be written
as (Sheikh et al. 2019)

$\dot{\nu}_{\min}=\frac{\Delta\nu_{\text{bin}}}{t_{\text{obs}}},$ (41)

where $\nu_{\text{bin}}$ is the size of the frequency bin, $t_{\text{obs}}$ is
the duration of the observation. To see tiny drift from the result, the drift
rate of the signal should be greater than $\dot{\nu}_{\min}$. That is, the
frequency resolution of the telescope should be grater than
$\Delta\nu_{\text{bin}}$.

Theoretically, if the observation period for an exoplanet is long enough, the
resulting ETI signal should completely or partly exhibit a pseudosinusoidal
curve (See Figure 7). However, the signal can only be received when we are in
the hemisphere facing the observation target. The final pseudosinusoidal
curves are likely to be intermittent (See Figure 11), unless the orbital
periods of exoplanets are shorter than the observation period.

Figure 11: Intermittent pseudosinusoidal curves of simulated long-term
observations for Kepler-438 b, TRAPPIST-1 g and Proxima Centauri b. Every
observation lasts for approximately four hours.

What we expect to see is the peak or trough parts of the curves, since the
other parts still appear linear characteristics. In such cases, if the
observation period $\tau_{\text{obs}}$ is shorter than a quarter of the period
of motion $P$, it should coincide with the phase of peak or trough part (See
figure 12), and the corresponding probability is

$p=\frac{4\tau_{\text{obs}}}{P}.$ (42)

The longer the observation time is, the greater the probability will be.

Figure 12: The situation that observation time $\tau_{\text{obs}}$ coincides
with the phase of peak or trough part. For a pseudosinusoidal curve, the
monotonicity changes in the peak or trough part.

And in the case in which the observation period $\tau_{\text{obs}}$ is longer
than a quarter of the period of motion $P$, or even longer than $P$, just as
Figure 11 shows, we are likely to observe such intermittent signal curves, or
the change of monotonicity in the peak or trough parts if the signal comes
from an exoplanet. The pseudosinusoidal curve criterion and the idea of
prolonging observation period will not be applied to the first observation,
since it is usually a survey for plenty of targets with a shorter observation
period. If the observation period $\tau_{\text{obs}}$ is long enough, we
recommend arranging an amount of observing period in the follow-up targeted
observation to determine potential ETI signals or signals of interest detected
in the survey. If there is no signal detected in the survey or the observation
time is relatively short, it is not necessary to spend such a long time for
observation. We also recommend combining other science goals to increase the
efficiency of the telescope resources since the probability to detect ETI
signal is very low. It can also increase the possibility of detecting other
astrophysical phenomena from the observation.

Orbits with higher eccentricities can make ETI signals more distinguishable
owing to the asymmetry of the frequency variations. The accelerations are
bigger when approaching the periapsis, while they get smaller when approaching
the apoapsis. Such an inequality of accelerations is reflected in the
asymmetry of frequency variations. And it may also be distinguished from RFI,
especially from common narrowband RFI, which appear to be symmetric over time.

##  5 Conclusion

For the commonly studied case, the drift rate of the signal is caused by the
rotational and orbital movements of Earth and the exoplanet. The drift rate
caused by the rotations of Earth and exoplanet can be regarded as uniform
circular motion. And the drift rate caused by the orbits of Earth and
exoplanet can be regarded as elliptical motion. In the previous studies, the
orbital motions were considered as uniform. However, the angular velocity, as
the equations derived in Section 2 illustrate, is nonuniform in an elliptical
orbital motion. Such nonuniform orbit motion can be explained by the Kepler
equation, and the Bessel’s solution of Kepler equation can be extended into
Fourier sine series. Apart from the celestial mechanics factors, The latitude
of telescope, the decl. and the ecliptic latitude of the target, and the
inclination of the exoplanet should be also taken into consideration.

Then, the equations derived are used in the calculations for the cases of
potential habitable planets: TRAPPIST-1 g, Kepler-438 b and Proxima Centauri
b, and FAST is the receiver on the Earth. We also discuss the case for
interferometers, signals from GC, RFI, spacecraft, space probes and
interstellar probes in Section 3. Finally, a discussion on the effects of
observation periods, orbital periods, and the eccentricities of exoplanets is
carried out in Section 4.

Based on Section 3 and Section 4 we can conclude the following:

  1. 1.

ETI signals can be distinguished by the intermittent and asymmetric
pseudosinusoidal curve of the frequency drift. The intermittency of the curve
is the fact that the targets can be only detected when we face them, and the
asymmetry of the curve is caused by the high eccentricities of the exoplanets’
orbits.

  2. 2.

Longer observation time and shorter orbital periods of exoplanets can increase
the possibility of seeing the intermittent curves, especially the peak or
trough parts, or even the complete pseudosinusoidal curves.

  3. 3.

This method can also be applied to the observations of interferometers in the
future, and the phase difference of the signal detected by different
telescopes can be a easy way to distinguish ETI signals.

  4. 4.

Long-term targeted observation can be arranged if there are potential ETI
signals or signals of interest found in the SETI survey. Including other
science goals can enhance the efficiency of the telescope and increase the
possibility for more science outcomes.

We thank the anonymous referees for the comments that helped us greatly
improve this article. And thank Dan Werthimer for his great suggestions. This
work was supported by the supported by the National Key R & D Program of China
under Grant No. 2018YFA0404204 and National Science Foundation of China under
Grants No. 11929301.

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