A
binary star is a
star
system consisting of two
stars orbiting around their common
center of mass. The brighter star is called
the
primary and the other is its
companion
star,
comes, or
secondary. Research between the early 1800s and
today suggests that many stars are part of either binary star
systems or star systems with more than two stars, called
multiple star
systems. The term
double star may be used
synonymously with
binary star, but more generally, a
double star may be either a binary star
or an
optical double star which consists of two stars with
no physical connection but which appear close together in the sky
as seen from the
Earth. A double star may be
determined to be optical if its components have sufficiently
different
proper motions or
radial velocities, or if
parallax measurements reveal its two components to
be at sufficiently different distances from the Earth. Most known
double stars have not yet been determined to be either bound binary
star systems or optical doubles.
Binary star systems are very important in
astrophysics because calculations of their
orbits allow the
masses of their component
stars to be directly determined, which in turn allows other stellar
parameters, such as radius and density, to be indirectly estimated.
This also determines an empirical mass-luminosity relationship
(MLR) from which the masses of single stars can be estimated.
Binary stars are often detected optically, in which case they are
called
visual binaries. Many visual binaries have long
orbital periods of several centuries or millennia and therefore
have orbits which are uncertain or poorly known. They may also be
detected by indirect techniques, such as
spectroscopy (
spectroscopic binaries)
or
astrometry (
astrometric
binaries). If a binary star happens to orbit in a plane along
our line of sight, its components will mutually
eclipse and
transit each other; these pairs are
called
eclipsing binaries, or, as they are detected by
their changes in brightness during eclipses and transits,
photometric binaries.
If the orbits of components in binary star systems are close enough
they can gravitationally distort their mutual outer stellar
atmospheres. In some cases, these
close binary systems can
exchange mass, which may bring their
evolution to stages that single stars
cannot attain. Examples of binaries are
Algol
(an eclipsing binary),
Sirius, and
Cygnus X-1 (of which one member is probably a
black hole). Binary stars are also common
as the nuclei of many
planetary
nebulae, and are the progenitors of both
novae and
type Ia
supernovae.
Discovery
The term
binary was first used in this context by Sir
William Herschel in 1802, when he
wrote:
- "If, on the contrary, two stars should really be situated
very near each other, and at the same time so far insulated as not
to be materially affected by the attractions of neighbouring stars,
they will then compose a separate system, and remain united by the
bond of their own mutual gravitation towards each other.
This should be called a real double star; and any two stars
that are thus mutually connected, form the binary sidereal system
which we are now to consider."
By the modern definition, the term
binary star is
generally restricted to pairs of stars which revolve around a
common centre of mass. Binary stars which can be
resolved with a telescope or
interferometric methods are known as
visual binaries. Most of the known visual binary stars
have not completed one whole revolution, but are observed to have
travelled along a curved path or a partial arc.
The more general term
double
star is used for pairs of stars which are seen to be close
together in the sky. This distinction is rarely made in languages
other than
English.
Double stars may be binary systems or may be
merely two stars that appear to be close together in the sky but
have vastly different true distances from the Sun. The latter are
termed
optical doubles or
optical pairs.
Since the invention of the
telescope, many
pairs of
double stars have been found.
Early examples include
Mizar and
Acrux. Mizar, in the
Big Dipper (
Ursa
Major), was observed to be double by
Giovanni Battista Riccioli in
1650 (and probably earlier by
Benedetto Castelli and
Galileo). The bright southern star
Acrux, in the
Southern
Cross, was discovered to be double by Father Fontenay in
1685.
John Michell was the first to suggest
that double stars might be physically attached to each other when
he argued in 1767 that the probability that a double star was due
to a chance alignment was small.
William Herschel began observing double
stars in 1779 and soon thereafter published catalogs of about 700
double stars. By 1803, he had observed changes in the relative
positions in a number of double stars over the course of 25 years,
and concluded that they must be binary systems; the first
orbit of a binary star, however, was not computed
until 1827, when
Félix Savary
computed the orbit of
Xi Ursae
Majoris. Since this time, many more double stars have been
catalogued and measured.
The Washington Double Star
Catalog, a database of visual double
stars compiled by the United States
Naval Observatory
, contains over 100,000 pairs of double stars, including optical doubles as well
as binary stars. Orbits are known for only a few thousand of
these double stars, and most have not been ascertained to be either
true binaries or optical double stars. This can be determined by
observing the relative motion of the pairs. If the motion is part
of an
orbit, or if the stars have similar
radial velocities and the
difference in their
proper motions is
small compared to their common proper motion, the pair is probably
physical. One of the tasks that remains for visual observers of
double stars is to obtain sufficient observations to prove or
disprove gravitational connection.
Classifications
By methods of observation
Binary stars are classified into four types according to the way in
which they are observed: visually, by observation;
spectroscopically, by periodic changes in
spectral lines;
photometrically, by changes in brightness
caused by an eclipse; or
astrometrically, by measuring a deviation in
a star's position caused by an unseen companion. Any binary star
can belong to several of these classes; for example, several
spectroscopic binaries are also eclipsing binaries.
Visual binaries
A
visual binary star is a binary star
for which the angular separation between the two components is
great enough to permit them to be observed as a
double star in a
telescope. The
resolving power of the telescope is an
important factor in the detection of visual binaries, and as
telescopes become larger and more powerful an increasing number of
visual binaries will be detected. The brightness of the two stars
is also an important factor, as brighter stars are harder to
separate, due to their glare, than dimmer ones are.
The brighter star of a visual binary is the
primary star,
and the dimmer is considered the
secondary. In some
publications (especially older ones), a faint secondary is called
the
comes (plural
comites;
.) If the stars are the same brightness, the discoverer designation
for the primary is customarily accepted.
The
position angle of the secondary
with respect to the primary is measured, together with the angular
distance between the two stars. The time of observation is also
recorded. After a sufficient number of observations are recorded
over a period of time, they are plotted in
polar coordinate with the primary
star at the origin, and the most probable
ellipse is drawn through these points such that the
Keplerian law of
areas is satisfied. This ellipse is known as the
apparent
ellipse, and is the projection of the actual elliptical
orbit of the secondary with respect to the
primary on the plane of the sky. From this projected ellipse the
complete elements of the orbit may be computed, with the
semi-major axis being expressed in angular
units unless the
stellar parallax, and
hence the distance, of the system is known.
Spectroscopic binaries
Sometimes, the only evidence of a binary star comes from the
Doppler effect on its emitted
light. In these cases, the binary consists of
a pair of stars where the
spectral
lines in the light emitted from each star shifts first toward
the blue, then toward the red, as each moves first toward us, and
then away from us, during its motion about their common
center of mass, with the period of their
common orbit.
In these systems, the separation between the stars is usually very
small, and the
orbital velocity very high.
Unless the plane of the orbit happens to be perpendicular to the
line of sight, the orbital velocities will have components in the
line of sight and the observed
radial
velocity of the system will vary periodically. Since radial
velocity can be measured with a
spectrometer by observing the
Doppler shift of the stars'
spectral lines, the binaries detected in this
manner are known as
spectroscopic binaries. Most of these
cannot be resolved as a visual binary, even with
telescopes of the highest existing
resolving power.
In some spectroscopic binaries, spectral lines from both stars are
visible and the lines are alternately double and single. Such a
system is known as a double-lined spectroscopic binary (often
denoted "SB2"). In other systems, the spectrum of only one of the
stars is seen and the lines in the spectrum shift periodically
towards the blue, then towards red and back again. Such stars are
known as single-lined spectroscopic binaries ("SB1").
The
orbit of a spectroscopic binary is
determined by making a long series of observations of the radial
velocity of one or both components of the system. The observations
are plotted against time, and from the resulting curve a period is
determined. If the orbit is
circular then the
curve will be a
sine curve.
If the orbit is
elliptical, the shape of the
curve will depend on the
eccentricity of the ellipse and the
orientation of the major axis with reference to the line of
sight.
It is impossible to determine individually the
semi-major axis a and the
inclination of the orbit plane
i. However, the product of
the semi-major axis and the sine of the inclination (i.e.
a sin
i) may be determined directly in linear
units (e.g. kilometres). If either
a or
i can be
determined by other means, as in the case of eclipsing binaries, a
complete solution for the orbit can be found.
Binary stars that are both visual and spectroscopic binaries are
rare, and are a precious source of valuable information when found.
Visual binary stars often have large true separations, with periods
measured in decades to centuries; consequently, they usually have
orbital speeds too small to be measured spectroscopically.
Conversely, spectroscopic binary stars move fast in their orbits
because they are close together, usually too close to be detected
as visual binaries. Binaries that are both visual and spectroscopic
thus must be relatively close to Earth.
Eclipsing binaries
An eclipsing binary, with an
indication of the variation in intensity.
An
eclipsing binary star is a binary star in which the
orbit plane of the two
stars lies so nearly in the line of sight of the
observer that the components undergo mutual
eclipses. In the case where the binary is also a
spectroscopic binary and the
parallax of
the system is known, the binary is quite valuable for stellar
analysis.
Algol is the best-known example of
an eclipsing binary.
In the last decade, measurement of eclipsing binaries' fundamental
parameters has become possible with 8 meter class telescopes. This
makes it feasible to use them as
standard candles.
Recently, they have been used to give direct distance estimates to
the
LMC,
SMC,
Andromeda Galaxy and
Triangulum Galaxy. Eclipsing binaries
offer a direct method to gauge the distance to galaxies to a new
improved 5% level of accuracy.
Eclipsing binaries are
variable stars,
not because the light of the individual components vary but because
of the eclipses. The
light curve of an
eclipsing binary is characterized by periods of practically
constant light, with periodic drops in intensity. If one of the
stars is larger than the other, one will be obscured by a total
eclipse while the other will be obscured by an
annular eclipse.
The period of the
orbit of an
eclipsing binary may be determined from a study of the light curve,
and the relative sizes of the individual stars can be determined in
terms of the radius of the orbit by observing how quickly the
brightness changes as the disc of the near star slides over the
disc of the distant star. If it is also a spectroscopic binary the
orbital elements can also be
determined, and the mass of the stars can be determined relatively
easily, which means that the relative densities of the stars can be
determined in this case.
Astrometric binaries
Astronomers have discovered some stars that seemingly orbit around
an empty space.
Astrometric binaries are relatively nearby
stars which can be seen to wobble around a point in space, with no
visible companion. The same mathematics used for ordinary binaries
can be applied to infer the
mass of the missing
companion. The companion could be very dim, so that it is currently
undetectable or masked by the glare of its primary, or it could be
an object that emits little or no
electromagnetic radiation, for
example a
neutron star.
The visible star's position is carefully measured and detected to
vary, due to the gravitational influence from its counterpart. The
position of the star is repeatedly measured relative to more
distant stars, and then checked for periodic shifts in position.
Typically this type of measurement can only be performed on nearby
stars, such as those within 10
parsecs.
Nearby stars often have a relatively high
proper motion, so astrometric binaries will
appear to follow a
sinusoidal path across
the sky.
If the companion is sufficiently massive to cause an observable
shift in position of the star, then its presence can be deduced.
From precise
astrometric measurements of
the movement of the visible star over a sufficiently long period of
time, information about the mass of the companion and its orbital
period can be determined. Even though the companion is not visible,
the characteristics of the system can be determined from the
observations using
Kepler's
law.
This method of detecting binaries is also
used to
locate extrasolar planets
orbiting a star. However, the requirements to perform this
measurement are very exacting, due to the great difference in the
mass ratio, and the typically long period of the planet's orbit.
Detection of position shifts of a star is a very exacting science,
and it is difficult to achieve the necessary precision. Space
telescopes can avoid the bluring effect of the
Earth's atmosphere, resulting in more
precise resolution.
By configuration of the system
Another classification is based on the distance of the stars,
relative to their sizes:
Detached binaries are binary stars where each component is
within its
Roche lobe, i.e. the area
where the
gravitational pull of the star
itself is larger than that of the other component. The stars have
no major effect on each other, and essentially evolve separately.
Most binaries belong to this class.
Semidetached binary stars are binary stars where one of
the components fills the binary star's Roche lobe and the other
does not. Gas from the surface of the Roche lobe filling component
(donor) is transferred to the other, accreting star. The
mass transfer dominates the evolution of the
system. In many cases, the inflowing gas forms an
accretion disc around the accretor.
A
contact binary is a type
of binary star in which both components of the binary fill their
Roche lobes. The uppermost part of the
stellar atmospheres forms a
common
envelope that surrounds both stars. As the friction of the
envelope brakes the
orbital motion,
the stars may eventually merge.
Cataclysmic variables and X-ray binaries
When a binary system contains a
compact
object such as a
white dwarf,
neutron star or
black hole, gas from the other,
donor, star can
accrete onto
the compact object. This releases
gravitational potential
energy, causing the gas to become hotter and emit radiation.
Cataclysmic variables, where
the compact object is a white dwarf, are examples of such
systems.In
X-ray binaries, the
compact object can be either a
neutron
star or a
black hole.
These binaries are classified as
low-mass or
high-mass according to the mass of
the donor star. High-mass X-ray binaries contain a young, early
type, high-mass donor star which transfers mass by its
stellar wind, while low-mass X-ray binaries are
semidetached binaries in which gas from a late-type donor star
overflows the Roche lobe and falls towards the neutron star or
black hole. Probably the best known example of an X-ray binary at
present is the
high-mass X-ray
binary Cygnus X-1. In Cygnus X-1, the
mass of the unseen companion is believed to be about nine times
that of our sun, far exceeding the
Tolman-Oppenheimer-Volkoff
limit for the maximum theoretical mass of a neutron star. It is
therefore believed to be a black hole; it was the first object for
which this was widely believed.
Designations
The components of binary stars are denoted by the suffixes
A and
B appended to the system's designation,
A denoting the primary and
B the secondary. The
suffix
AB may be used to denote the pair (for example, the
binary star α Centauri AB consists of the stars α Centauri A and α
Centauri B.) Additional letters, such as
C,
D,
etc., may be used for systems with more than two stars. In cases
where the binary star has a
Bayer
designation and is widely separated, it is possible that the
members of the pair will be designated with superscripts; an
example is
ζ Reticuli, whose
components are ζ
1 Reticuli and ζ
2
Reticuli.
Double stars are also designated by an abbreviation giving the
discoverer together with an index number. α Centauri, for example,
was found to be double by Father Richaud in 1689, and so is
designated
RHD 1. These discoverer codes can be found in
the
Washington Double
Star Catalog.
Binary star evolution
Formation
While it is not impossible that some binaries might be created
through
gravitational capture between
two single stars, given the very low likelihood of such an event
(three objects are actually required, as conservation of energy
rules out a single gravitating body capturing another) and the high
number of binaries, this cannot be the primary formation process.
Also, the observation of binaries consisting of pre
main sequence stars, supports the theory that
binaries are already formed during
star
formation. Fragmentation of the molecular cloud during the
formation of
protostars is an acceptable
explanation for the formation of a binary or multiple star
system.
The outcome of the
three body
problem, where the three stars are of comparable mass, is that
eventually one of the three stars will be ejected from the system
and, assuming no significant further perturbations, the remaining
two will form a stable binary system.
Mass transfer and accretion
As a
main sequence star increases in
size during its
evolution, it may
at some point exceed its
Roche lobe,
meaning that some of its matter ventures into a region where the
gravitational pull of its companion star
is larger than its own. The result is that matter will transfer
from one star to another through a process known as Roche Lobe
overflow (RLOF), either being absorbed by direct impact or through
an
accretion disc. The
mathematical point through which this transfer
happens is called the first
Lagrangian
point It is not uncommon that the accretion disc is the
brightest (and thus sometimes the only visible) element of a binary
star.
An animation of an eclipsing binary
system undergoing mass transfer.
If a star grows outside of its Roche lobe too fast for all abundant
matter to be transferred to the other
component, it is also possible that matter will leave the system
through other Lagrange points or as
stellar
wind, thus being effectively lost to both components.Since the
evolution of a star is determined by its mass, the process
influences the evolution of both companions, and creates stages
that can not be attained by single
stars.
Studies of the eclipsing ternary
Algol led to
the
Algol paradox in the
theory of
stellar evolution:
although components of a binary star form at the same time, and
massive stars evolve much faster than the less massive ones, it was
observed that the more massive component Algol A is still in the
main sequence, while the less massive
Algol B is a
subgiant star at a later
evolutionary stage. The paradox can be solved by
mass transfer: when the more massive star
became a subgiant, it filled its
Roche
lobe, and most of the mass was transferred to the other star,
which is still in the main sequence. In some binaries similar to
Algol, a gas flow can actually be seen.
Runaways and novae
It is also possible for widely separated binaries to lose
gravitational contact with each other during their lifetime, as a
result of external perturbations. The components will then move on
to evolve as single stars. A close encounter between two binary
systems can also result in the gravitational disruption of both
systems, with some of the stars being ejected at high velocities,
leading to
runaway stars.
If a
white dwarf has a close companion
star that overflows its
Roche lobe, the
white dwarf will steadily accrete gases from the star's outer
atmosphere. These are compacted on the white dwarf's surface by its
intense gravity, compressed and heated to very high temperatures as
additional material is drawn in. The white dwarf consists of
degenerate matter, and so is
largely unresponsive to heat, while the accreted hydrogen is not.
Hydrogen fusion can occur in a stable
manner on the surface through the
CNO
cycle, causing the enormous amount of energy liberated by this
process to blow the remaining gases away from the white dwarf's
surface. The result is an extremely bright outburst of light, known
as a
nova.
In extreme cases this event can cause the white dwarf to exceed the
Chandrasekhar limit and trigger
a
supernova that destroys the entire star,
and is another possible cause for runaways. A famous example of
such an event is the supernova
SN 1572,
which was observed by
Tycho Brahe. The
Hubble Space Telescope
recently took a picture of the remnants of this event.
Use in astrophysics
Binaries provide the best method for
astronomers to determine the mass of a distant
star. The gravitational pull between them causes them to orbit
around their common center of mass. From the orbital pattern of a
visual binary, or the time variation of the spectrum of a
spectroscopic binary, the mass of its stars can be determined. In
this way, the relation between a star's appearance (
temperature and
radius)
and its mass can be found, which allows for the determination of
the mass of non-binaries.
Because a large proportion of stars exist in binary systems,
binaries are particularly important to our understanding of the
processes by which stars form. In particular, the period and masses
of the binary tell us about the amount of
angular momentum in the system. Because
this is a
conserved quantity in
physics, binaries give us important clues
about the conditions under which the stars were formed.
Research findings
It is estimated that approximately 1/3 of the
star systems in the
Milky
Way are binary or multiple, with the remaining 2/3 consisting
of single
stars.
There is a direct correlation between the
period of
revolution of a binary star and the
eccentricity of its orbit, with systems
of short period having smaller eccentricity. Binary stars may be
found with any conceivable separation, from pairs orbiting so
closely that they are practically in contact with each other, to
pairs so distantly separated that their connection is indicated
only by their common
proper motion
through space. Among gravitationally bound binary star systems,
there exists a so called
log
normal distribution of periods, with the majority of these
systems orbiting with a period of about 100 years. This is
supporting evidence for the theory that binary systems are formed
during
star formation.
In pairs where the two stars are of equal
brightness, they are also of the same
spectral type.In systems
where the brightnesses are different, the fainter star is bluer if
the brighter star is a
giant star, and
redder if the brighter star belongs to the
main sequence.
The mass of a star can be directly determined only from its
gravitational attraction. Apart from the
Sun and
stars which act as
gravitational
lenses, this can be done only in binary and multiple star
systems, making the binary stars an important class of stars. In
the case of a visual binary star, after the orbit and the
stellar parallax of the system has been determined,
the combined mass of the two stars may be obtained by a direct
application of the
Keplerian harmonic
law.
Unfortunately, it is impossible to obtain the complete orbit of a
spectroscopic binary unless it is also a visual or an eclipsing
binary, so from these objects only a determination of the joint
product of mass and the
sine
of the angle of inclination relative to the line of sight is
possible. In the case of eclipsing binaries which are also
spectroscopic binaries, it is possible to find a complete solution
for the specifications (mass,
density, size,
luminosity, and approximate shape) of
both members of the system.
Planets around binary stars
Science fiction has often featured
planets of binary or ternary stars as a
setting. In reality, some orbital ranges are impossible for
dynamical reasons (the planet would be expelled from its orbit
relatively quickly, being either ejected from the system altogether
or transferred to a more inner or outer orbital range), whilst
other orbits present serious challenges for eventual
biospheres because of likely extreme variations in
surface temperature during different parts of the orbit. Planets
that orbit just one star in a binary pair are said to have "S-type"
orbits, whereas those that orbit around both stars have "P-type" or
"
circumbinary" orbits. It is
estimated that 50–60% of binary stars are capable of supporting
habitable terrestrial planets within stable orbital ranges.
Simulations have shown that the presence of a binary companion can
actually improve the rate of planet formation within stable orbital
zones by "stirring up" the protoplanetary disk, increasing the
accretion rate of the protoplanets within.
Detecting planets in multiple star systems introduces additional
technical difficulties, which may be why they are only rarely
found. Examples include
PSR B1620-26
b and
Gamma Cephei.
A study of fourteen previously known planetary systems found three
of these systems to be binary systems. All planets were found to be
in S-type orbits around the primary star. In these three cases the
secondary star was much dimmer than the primary and so was not
previously detected. This discovery resulted in a recalculation of
parameters for both the planet and the primary star.
Binary star examples
The large distance between the components, as well as their
difference in color, make
Albireo one of the
easiest observable visual binaries. The brightest member, which is
the third brightest star in the
constellation Cygnus, is actually a close binary
itself. Also in the Cygnus constellation is
Cygnus X-1, an
X-ray source
considered to be a
black hole. It is a
high-mass X-ray binary, with
the optical counterpart being a
variable
star. Another famous binary is
Sirius,
the brightest star in the night time sky, with a visual
apparent magnitude of −1.46. It is
located in the constellation
Canis
Major. In 1844
Friedrich Bessel
deduced that Sirius was a binary. In 1862
Alvan Graham Clark discovered the
companion (Sirius B; the visible star is Sirius A).
In 1915 astronomers at
the Mount Wilson
Observatory
determined that Sirius B was a white dwarf, the first to be discovered.
In 2005, using the
Hubble Space
Telescope, astronomers determined Sirius B to be in diameter,
with a mass that is 98% of the
Sun.
An example of an eclipsing binary is
Epsilon Aurigae in the constellation
Auriga. The visible component
belongs to the
spectral class
F0, the other (eclipsing) component is not visible. The next such
eclipse occurs from 2009–2011, and it is hoped that the extensive
observations that will likely be carried out may yield further
insights into the nature of this system. Another eclipsing binary
is
Beta Lyrae, which is a semi-detached
binary star system in the constellation of
Lyra.
Other interesting binaries include
61 Cygni
(a binary in the constellation
Cygnus, composed of two
K class main
sequence stars, 61 Cygni A and 61 Cygni B, which is known for
its large
proper motion),
Procyon (the brightest star in the constellation
Canis Minor and the eighth brightest
star in the night time sky, which is a binary consisting of the
main star with a faint
white dwarf
companion), SS Lacertae (an eclipsing binary which stopped
eclipsing), V907 Sco (an eclipsing binary which stopped, restarted,
then stopped again) and
BG Geminorum
(an eclipsing binary which is thought to contain a black hole with
a K0 star in orbit around it).
Multiple star examples
Systems with more than two stars are termed
multiple stars.
Algol is
the most famous ternary (long thought to be a binary), located in
the constellation
Perseus.
Two components of the system eclipse each other, the variation in
the intensity of Algol first being recorded in 1670 by
Geminiano Montanari. The name Algol
means "demon star" (from
Arabic
الغول al-ghūl),
which was probably given due to its peculiar behavior. Another
visible ternary is
Alpha Centauri, in
the southern constellation of
Centaurus,
which contains the
fourth
brightest star in the night sky, with an
apparent visual magnitude of −0.01. This
system also underscores the fact that binaries need not be
discounted in the search for habitable planets. Alpha Centauri A
and B have an 11 AU distance at closest approach, and both
should have stable habitable zones.
There are also examples of systems beyond ternaries:
Castor is a sextuple star system, which is the
second brightest star in the constellation
Gemini and one of the brightest stars
in the nighttime sky. Astronomically, Castor was discovered to be a
visual binary in 1719. Each of the components of Castor is itself a
spectroscopic binary. Castor also has a faint and widely separated
companion, which is also a spectroscopic binary.
See also
Notes and references
- Pronounced
- p. 481, Catalogue of 500 New Nebulae, Nebulous Stars, Planetary
Nebulae, and Clusters of Stars; With Remarks on the Construction of
the Heavens, William Herschel, Philosophical Transactions
of the Royal Society of London,92 (1802), pp.
477–528.
- The Binary Stars, Robert Grant Aitken, New York:
Dover, 1964, p. ix.
- The Binary Stars, Robert Grant Aitken, New York:
Dover, 1964, p. 1.
- Vol. 1, part 1, p. 422, Almagestum Novum,
Giovanni Battista Riccioli, Bononiae: Ex typographia haeredis
Victorij Benatij, 1651.
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