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\DOI{10.5802/crphys.153}
\datereceived{2023-03-20}
\dateaccepted{2023-03-28}
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\dateposted{2023-06-08}
\begin{document}

\begin{noXML}

%\TopicEF{Exoplanets}{Exoplan\`etes}

\title{Doppler cross-correlation spectroscopy as a path to the
detection of Earth-like planets} 

\alttitle{Spectroscopie Doppler par corr\'{e}lation crois\'{e}e : une
possibilit\'{e} pour d\'{e}tecter des plan\`{e}tes similaires \`{a} la
Terre}

\author{\firstname{Michel} \lastname{Mayor}\CDRorcid{0000-0002-9352-5935}}

\address{Astronomy Department, University of Geneva,
Ch.Pegasi 51, CH-1270 Versoix, Switzerland}
\email{michel.mayor@unige.ch}

\def\thanksname{Note}
\thanks{Follows up on a conference-debate of the French Academy of
Sciences entitled ``Exoplanets: the new challenges'' held on 18 May
2021, visible via
\url{https://www.academie-sciences.fr/fr/Colloques-conferences-et-debats/exoplanetes.html}}

\def\thanksname{Note}
\thanks{\textbf{Note.}  Fait suite \`{a} une conf\'{e}rence-d\'{e}bat
de l'Acad\'{e}mie des sciences intitul\'{e}e \guillemotleft{}~Exoplan\`{e}tes : les
nouveaux d\'{e}fis~\guillemotright{} tenue le 18 mai 2021, visible via
\url{https://www.academie-sciences.fr/fr/Colloques-conferences-et-debats/exoplanetes.html}}

%\editornote{Fait suite \`{a} une conf\'{e}rence-d\'{e}bat
%de l'Acad\'{e}mie des sciences intitul\'{e}e \guillemotleft{}~Exoplan\`{e}tes : les nouveaux d\'{e}fis~\guillemotright{} tenue le 18 mai 2021, visible via \url{https://www.academie-sciences.fr/fr/Colloques-conferences-et-debats/exoplanetes.html}}

\begin{abstract} 
In the middle of the 20th century, a paradigm shift appeared concerning
the expected frequency of planetary systems in the galaxy\,\ldots a shift
induced by the observation of the rotational velocities of low main
sequence stars (Struve 1952)! 

At the same time, Fellgett (1955) proposed to concentrate the diluted
Doppler information on several tens of thousands of absorption lines to
allow the precise measurement of stellar velocities. This idea improved
the efficiency of radial velocity measurements by a factor of over
1000. Gradually the accuracy of the new generation of spectrographs
using cross-correlation is improved from 300 m/s to 0.1 m/s.\,\ldots An
idea that will contribute in an important way to the discovery of 51
Pegasi b and several hundreds of planetary systems.

Will visible or infrared cross-correlation spectrographs today be able
to detect rocky planets in the habitable zone associated with their
host star?
\end{abstract}

\begin{altabstract} 
Au milieu du 20\`{e}me si\`{e}cle, un changement de paradigme est
apparu concernant la fr\'{e}quence attendue des syst\`{e}mes
plan\'{e}taires dans la galaxie\,\ldots un changement induit par
l'observation des vitesses de rotation des \'{e}toiles de la
s\'{e}quence principale basse (Struve 1952) !

A la m\^{e}me \'{e}poque, Fellgett (1955) propose de concentrer
l'information Doppler dilu\'{e}e sur plusieurs dizaines de milliers de
raies d'absorption pour permettre la mesure pr\'{e}cise des vitesses
stellaires. Cette id\'{e}e a permis d'am\'{e}liorer l'efficacit\'{e}
des mesures de vitesse radiale d'un facteur sup\'{e}rieur \`{a} 1000.
Progressivement, la pr\'{e}cision de la nouvelle g\'{e}n\'{e}ration de
spectrographes utilisant la corr\'{e}lation crois\'{e}e est
am\'{e}lior\'{e}e de 300 m/s \`{a} 0,1 m/s.\,\ldots Une id\'{e}e qui
contribuera de mani\`{e}re importante \`{a} la d\'{e}couverte de 51~Pegasi~b et de plusieurs centaines de syst\`{e}mes plan\'{e}taires.

Les spectrographes \`{a} corr\'{e}lation crois\'{e}e dans le visible
ou l'infrarouge seront-ils aujourd'hui capables de d\'{e}tecter des
plan\`{e}tes rocheuses dans la zone habitable associ\'{e}e \`{a} leur
\'{e}toile h\^{o}te ?
\end{altabstract}


\keywords{\kwd{Astronomy}
\kwd{Planetology}
\kwd{Exoplanets}
\kwd{Exobiology}
\kwd{Velocimetry}}


\altkeywords{\kwd{Astronomie}
\kwd{Plan\'{e}tologie}
\kwd{Exoplan\`{e}tes}
\kwd{Exobiologie}
\kwd{V\'{e}locim\'{e}trie}} 

\maketitle

\end{noXML}

{\vspace*{-1pc}}

\section{Change of paradigm during the 20th century} 

The last 25 years have seen this ancient dream of the ``plurality of
worlds'' transformed into an exciting area of 21st century astrophysics.

In a famous letter, Epicurus in the 4th century BC already wondered
about the possibility that some of these Worlds might contain living
species. The hypothesis of the plurality of Worlds, and even the
possibility that some of them shelter living species, has been present
in philosophical thinking over the last two millennia. 

However, the astronomers in the first half of the 20th century
estimated that our planetary system was unique among the hundreds of
billions of stars in our Galaxy. What was the origin of the nebulae
required to explain the formation of planets with coplanar orbits and
same direction of rotation? The erroneous assumption that the disk
originated from tidal effects resulting from a very close passage of
two stars (near-collision) was at the origin of these erroneous
estimates of the frequency of planetary systems. Indeed, the
probability of a near-collision, despite billions of years and hundreds
of billions of stars, is close to zero. 

The discovery in 1943 of planets hosted by stars in the solar vicinity
led to a complete change in the estimate of the number of planetary
systems\,\ldots for a wrong reason! These discoveries soon proved to be
wrong (Dick~\cite{1}).

Struve~\cite{2} noted that the rotations of low main sequence stars are
very small despite the large angular momentum expected from star
formation by gravitational collapse of turbulent gas:

\begin{quote}
\textit{``I have suggested elsewhere that the lack of rapid axial
rotation of normal solar-type stars\,{\ldots} suggests that these stars
have converted their axial rotation to angular momentum from the
orbital motion of the planets. Therefore, there can be many planet-like
objects in the galaxy''} (Struve~\cite{2}).
\end{quote}


Change of paradigm: the estimation of planetary systems in the galaxy
jumps to billions or hundred of billions (Figure~\ref{fig1}).



In the 1970s, the detection of strong infrared emission from very young
stars revealed the presence of protoplanetary disks (see for example
Beckwith~\cite{3}). 


In 1995, the high angular resolution of the Hubble Space Telescope
allowed McCaughrean and O'Dell~\cite{4} to obtain images of
these disks surrounding most of the young stars in the Orion 
Nebula (Figure~\ref{fig2}a).



And, more recently, the ALMA submillimeter interferometer has provided
astonishing images of the accretion disks associated with very young
T-Tauri stars (Figure~\ref{fig2}b).

The structure of these disks, which exhibit dark rings, suggests the
formation of planetary systems. 

\begin{quote}
\textit{There is no longer any doubt that most stars must be hosts to
multi-planetary systems: how can they be detected?}
\end{quote}

\section{Doppler spectroscopy}

The ratio of stellar luminosity to the luminosity reflected by a
possible planet made direct imaging beyond the possibilities of the
instrumentation of the time. Indirect detection seemed more promising.
Beloryzky~\cite{6} and Struve~\cite{2} had already mentioned the
possibility of detecting the existence of planets by measuring the
gravitational pull of the planet. This possibility was obviously beyond
the accuracy of the spectrographs of that time. The years 1970--1990 saw a
wide variety of instrumental developments to improve the sensitivity
and stability of spectrographs. I will only mention here the approaches
that led to exoplanet search programmes. On more than ten years,
Campbell and Walker~\cite{7} using an HF calibration cell carefully
measure the radial velocity of about 20 stars. Marcy and
Butler~\cite{8} made a survey of a larger stellar sample using an
iodine cell as calibration device. Baranne \etal~\cite{9,10}
developped cross-correlation Doppler spectrographs as the ELODIE
spectrograph, Baranne \etal~\cite{11}, installed on the 1.93 m
telescope at Haute-Provence Observatory.

\begin{figure}[!h]
\includegraphics{fig01}
\caption{\label{fig1} Estimated number of planetary systems in the
Milky Way during the 20ith Century (adapted from
Dick~\cite{1}).}
\end{figure}

\section{The discovery of 51 Pegasi b}

But let's go back 27 years, to the beginning of this exciting phase of
first discoveries.

Given the success of our CORAVEL instrument (Baranne 
\etal~\cite{9}) Philippe V\'{e}ron, director of the Haute-Provence
Observatory, asked Andr\'{e} Baranne and myself to develop a
cross-correlation spectrograph to be installed at the Cassegrain focus
of the 1.93-m telescope. The technical possibilities of the 1980s
offered decisive advantages: large CCD detector allowing the spectrum
to be recorded and digitally correlated with a mask adapted to the
variations in temperature and atmospheric pressure, optical fibers to
improve the scrambling of the beams illuminating the optics, etc. In
1990 we obtained (with difficulty!) the financing of this instrument
and in April 1994 it was possible to begin our survey to search for
exoplanets. Our stellar sample consists of 142 solar stars selected to
be simple, during a survey of more than ten years with the CORAVEL
spectrograph (Duquennoy and Mayor~\cite{12}). Our periods of
observation on the 1.93 m telescope were limited to one week every two
months. Our radial velocity precision was better than 15 m/s.

Our survey aims to extend the mass function of companions orbiting
solar-type stars beyond the lower limit of hydrogen-burning (i.e., the
domain of brown dwarfs and planets). In January 1995, several stars
exhibit significant radial velocity variations, some with strong
chromospheric activity have non-periodic variations. However, a G2V
star appears to have a periodic velocity variation of only 4.2 days,
maybe resulting from a planetary companion with a minimum mass
(\textit{msini}) of about half the mass of Jupiter (estimated from
only 12 measurements).

\begin{figure}[!h]
{\vspace*{-3pt}}
\includegraphics{fig02}
{\vspace*{-3pt}}
\caption{\label{fig2}(a)~Accretion disks observed by the HST around
young stars of the Orion Nebula
(McCaughrean and O'Dell~\cite{4}). (b)~Disks substructure at High
Angular resolution measured with the ALMA submillimeter Interferometer
(Andrews \etal~\cite{5}).}
{\vspace*{-3pt}}
\end{figure}

The scenario of Jovian planet formation required the accretion of ice
particles (Boss~\cite{13}), thus orbital periods greater than 10 years.
Our planet candidate showed a disagreement of about a factor 1000 with
this formation scenario!

Periodic (or quasi-periodic) variations in stellar velocities can
result from pulsations, stellar rotation associated with surface
anisotropies induced by the stellar magnetic activity or due to the
influence of a companion.

Radial velocity variations only allow the determination of a minimum
mass (\textit{msini}). If the proper rotation of the star was 4.2 days,
one should have observed a very strong chromospheric activity, whereas
51 Pegasi does not show such signs. The probability of having a very
weak \textit{sini}, orbiting almost perpendicular to the line of sight,
is very low. The photometric stability was not consistent with a
pulsation. In addition, gravity modes (with $n=115$) do not pass through
the convective zone, etc.

The following season allowed us to verify the stability of the period,
the amplitude, and the phase of the radial velocity variation of 51
Pegasi: a test to exclude the hypothesis that the observed radial
velocity variation results from the influence of the magnetic activity
phenomena. The stability of the shape of the cross-correlation function
also helps to reject the possibility of pulsation (also rejected on
physical arguments and the observed photometric stability).

In July 1995, with Didier Queloz, at the Haute-Provence observatory, we
were convinced of the interpretation of the influence of a planet of
mass close to half a Jupiter. The announcement (Mayor and
Queloz~\cite{14}) was based on 34 measurements (Figure~\ref{fig3}).

\begin{figure}[t!]
{\vspace*{-3pt}}
\includegraphics{fig03}
{\vspace*{-4pt}}
\caption{\label{fig3}Measurements of the radial velocities of 51 Pegasi
b made in 1994 and 1995 at the Haute-Provence Observatory, Mayor and
Queloz~\cite{14}. The observations acquired at four different periods
attest of the stability of the variation.}
{\vspace*{-3pt}}
\end{figure}

The following year saw the announcement of the discovery of several
planets, often also with short periods, typical of what were then
called ``hot Jupiters''.

The origin of the short periods observed for these ``hot Jupiters'' was
published immediately after the announcement of the discovery of 51
Pegasi b (Lin \etal~\cite{15}). During the formation
of the planet, still immersed in the accretion disc, an angular
momentum transfer induces an orbital migration. In fact, this effect
had already been studied for 15 years\,\ldots but not considered by planet
researchers (Goldreich and Tremaine~\cite{16}, Papaloizou and
Lin~\cite{17}, Lin and Papaloizou~\cite{18}, Ward~\cite{19}). The
discovery of orbital migration through the discovery of short-period
planets is a major physical phenomenon for the understanding of
planetary system formation.

\section{The cross-correlation spectroscopy}

Several thousandth of absorption lines are present in the spectra of
cold stars. Figure~\ref{fig4} illustrates a limited range (about 10\%) of the
spectrum obtained with the ESPRESSO spectrograph, recently installed on
the VLT on Mount Paranal (ESO, Chile). The Doppler information is
diluted on this large number of absorption lines, each one being
determined by a limited number of photons. 

\begin{figure}[t!]
\includegraphics{fig04}
\caption{\label{fig4}10\% of the bandwidth used by the ESPRESSO
  spectrograph (Pepe \etal~\cite{22}). This figure
illustrates the richness of the Doppler information contained in the
spectrum of cool stars. But it is interesting to note that the Doppler
variation of the Sun induced by the gravitational influence of the
Earth. About 0.1 m/s corresponds to less than a thousandth of the width
of the thinnest of these spectral lines.}
\end{figure}

Fellgett's~\cite{20} proposal was to concentrate the Doppler
information by cross correlating a template typical of cold star
spectra with the observed spectrum. The position of the correlation
function provides the radial velocity derived from thousands of
absorption lines. 

A first correlation spectrograph by Griffin~\cite{21}
demonstrated the efficiency of this concept, reaching an accuracy of
500 m/s and an efficiency relative to the old photographic technique of
the order of a factor of 1000. 

The progress made in improving the sensitivity and stability of
cross-correlation spectrographs over the last 45 years has been
exceptional. 

Since 1971, I had the pleasure of collaborating with Andr\'{e} Baranne,
optics engineer, at the Marseille Observatory.

In 1977, our first instrument of this type, CORAVEL, allows a precision
for radial velocities of 300 m/s (Baranne \etal~\cite{9,10}),
then with ELODIE it was possible to detect stellar wobble with an
accuracy of 13 m/s in 1994 (Baranne \etal~\cite{11}), and for our third
generation of cross-correlation instrument, HARPS in 2003, allows
reaching 
1 m/s (Mayor \etal~\cite{23}). Finally, ESPRESSO in 2018 reaches the
exceptional accuracy of 
{0.1 m/s (Pepe \etal~}{~\cite{22}). Note that this exceptional precision
of 0.1 m/s corresponds approximately to the variation in the velocity
of the Sun due to the presence of the Earth.}
This improvement in accuracy is the result of numerous improvements to
the spectrograph
(vacuum instrument, octagonal fibres to improve mode mixing,
calibration unit with Fabry--Perot and lasercomb as calibration units,
etc). Improvements also results from multiple modifications of the
signal processing software (0.1 m/s corresponds to only one nanometer
in the detector plane!).

However, the real limit of accuracy is dominated by the variation of
the radial velocity induced by the various phenomena related to the
magnetic activity in the atmosphere of cold stars. Many efforts are
being made to reduce the influence of intrinsic stellar noise. Even for
stars with low magnetic activity, such as our Sun for example, a
variation in the measured radial velocity can be greater than one meter
per second. 


{Since 2015, a small telescope of about 10 cm feeds the HARPS
spectrograph installed on the 3.5 m Galileo telescope in La Palma
(Canary Islands) with the sun integrated light. A time series of
several years of the solar radial velocity has been obtained and allows
various signal processing approaches to be tested} {(Dumusque
\etal~\cite{24}).} {Signal processing using Gaussian processes
(Collier-Cameron \etal}{~\cite{25}) or using deep learning allows a
significant decrease in the rms (De~Beurs \etal~\cite{26}).}

Energy transport in the outer regions of the solar atmosphere results
from convective motions. Depending on their depth of formation in the
atmosphere, the absorption lines have a different sensitivity to the
variations in the speed of convection. These velocity variations are
themselves linked to variations in the magnetic field. Thus, a
cross-correlation analysis considering the depth of the lines
(Cretignier \etal~\cite{27,28}) allows a major decrease of the solar
radial velocity dispersion (Cretignier \etal~\cite{27}) and his PhD
thesis, Geneva University (2022).
Applied to the large number of stellar
spectra acquired with HARPS this approach reveals a very large number
of new planetary candidates with very small masses (Figure~\ref{fig5})
(the global rms of this large sample decreases from 1.3 m/s to 0.75
m/s). A large fraction of the remaining rms results from the photon
noise due to limited S/N of stellar measurements. We can expect a
significant increase of the S/N in the future by using the ESPRESSO
spectrograph connected to a VLT with a diameter of 8.2 m (and in some
cases connected to the four VLT on top of Paranal).

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig5} Minimum masses of planets detected by the radial
velocity method as function of the orbital period. Black symbols have
been used for the planetary candidates detected by the Yarara software
(Cretignier \etal~\cite{28}). The masses of host stars are coded by
different colors. A cross indicates the position of Earth.}
\end{figure}

Planets with minimum masses much smaller than the Earth-mass have
previously been detected. 

However, only for planets with quite short periods. The ESPRESSO
spectrograph was able to detect a short-period sub-Earth planet
orbiting Proxima Centauri (M $=$ 0.26 Earth-mass, Period $=$ 5.12 days,
semi-amplitude of the radial velocity variation $=$ 39~cm/s) (Faria
\etal~\cite{29}) (Figure~\ref{fig6}).

\begin{figure}[t!]
\includegraphics{fig06}
\caption{\label{fig6}Radial velocity variations of Proxima Centauri due
to its two closest planets (Faria \etal~\cite{29}).}
\end{figure}


The radial-velocity detection of Earth-twins in the habitable zone of
the one solar mass star is really challenging for solar-type stars.
However the habitable zone (HZ) is much closer to M-host stars. Several
planets have already been detected in the HZ of interesting low mass
stars of the solar neighborhood (Proxima Centauri b, our closest
neighbor (Anglada-Escud\'{e} \etal~\cite{31}); and seven planets hosted
by Trappist 1, a very low mass M-star with 3 planets in the HZ
(Gillion \mbox{\etal}~\cite{30}). 

\section{Earth-twins?}

Can we detect rocky planets in the habitable zone of a star like our Sun?

Let us first note that a good dozen planets with masses comparable to
that of our Earth have already been detected in the habitable zone of
their host star. Most of them are hosts of very small mass stars, for
which the habitable zone near the star corresponds to very short period
orbits. The most promising are Proxima Centauri (having a planet with
period 11 days, Anglada-Escud\'{e} \etal~\cite{31}) and 
Trappist-1~\cite{30}.

 This last star of mass close to the lower limit of the main sequence
is surrounded by seven rocky planets including three planets with a
temperature compatible with the presence of liquid water on their
surface (Gillon \etal~\cite{30}). These planets very close to their
host star are synchronized and thus always present the same hemisphere
facing the star. In addition, these planets are subjected to an intense
flux of energetic particles.

The luminosity received at the surface of these planets close to low
mass stars is in the infra-red. Do these conditions allow the
development of the complex chemistry involved in the development of
life? Perhaps, but it is also interesting to ask the question of the
detectability of rocky planets in the habitable zone of solar-type
stars or even slightly less massive (spectral type~K).

The detection of ``habitable'' planets is facilitated around low mass
stars because the amplitude of the variations of the radial velocity is
greater when the period is short and the mass of the star small. In
addition, the probability of detection by the planetary transit
technique also favors short periods.

These various factors do not act in favor of the detection of planets
hosted by K or G stars.

We have seen the difficulty to detect a Doppler variation of 0.08 m/s
hidden in the noise induced by the magnetic activity. The difficulty of
detecting a transit with a period of 6 months to a year and a contrast
of a few ten-thousandths is also delicate.

It is certain that the discovery of a planet through a transit will
help considerably to constrain the measurement of the planetary mass
required by cross-correlation spectroscopy. We can hope that the
various space missions currently in progress or expected soon (TESS,
CHEOPS, PLATO) may reveal some rocky planets in the habitable zone of
nearby K or G stars. 

\break
Such objects will offer choice targets for the study of their
atmosphere by transmission spectroscopy during transits (every 6 months
or year!). But the atmosphere of a planet of mass comparable to our
Earth is quite thin. The detection of its atmosphere and possibly of
biomarkers will require very high signal/noise spectra for such
measurements. The ESO telescope of 39~m \mbox{diameter} under construction on
Mount Armazones and its spectrograph ANDES (Marconi \mbox{\etal}~\cite{32})
will be adapted to such measurements.

The frequency of rocky planets orbiting in the habitable zone of stars
of spectral types G or K is not well constrained. Figures of the order
of a few percent have been mentioned in various studies. Let us note
that the surveys of stellar radial velocities 
(HARPS in the southern hemisphere and at HIRES at Keck for the north)
indicate a strongly increasing curve of the frequency of planets when
their mass decreases.

Let us also note that the numerical simulations of the planetary
formation indicate the massive presence of rocky planets of period
compatibles with their presence in the habitable zone (Benz
\etal~\cite{33}). 

\section*{Conflicts of interest}
Author has no conflict of interest to declare.

\section*{Acknowledgements}

I would like to dedicate these few pages to Roger Griffin and Andr\'{e}
Baranne who left us recently. This text highlights the role played by
these two colleagues in the development of correlation spectroscopy. 

From CORAVEL to ELODIE, HARPS and ESPRESSO, many friends, engineers,
technicians, and astrophysicists have participated in this exceptional
leap in precision from 300 m/s to 0.1 m/s, may they all be warmly
thanked.

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