Angle between the rotational axis and orbital axis of a body
In
astronomy
,
axial tilt
, also known as
obliquity
, is the
angle
between an object's
rotational axis
and its
orbital
axis, which is the line
perpendicular
to its
orbital plane
; equivalently, it is the angle between its
equatorial
plane and orbital plane.
[1]
It differs from
orbital inclination
.
At an obliquity of 0 degrees, the two axes point in the same direction; that is, the rotational axis is perpendicular to the orbital plane.
The rotational axis of
Earth
, for example, is the imaginary line that passes through both the
North Pole
and
South Pole
, whereas the Earth's orbital axis is the line perpendicular to the imaginary
plane
through which the Earth moves as it revolves around the
Sun
; the Earth's obliquity or axial tilt is the angle between these two lines.
Over the course of an
orbital period
, the obliquity usually does not change considerably, and the
orientation of the axis remains the same
relative to the
background
of
stars
. This causes one pole to be pointed more toward the Sun on one side of the orbit, and more away from the Sun on the other side?the cause of the
seasons
on Earth.
Standards
[
edit
]
There are two standard methods of specifying a planet's tilt. One way is based on the planet's
north pole
, defined in relation to the direction of Earth's north pole, and the other way is based on the planet's
positive pole
, defined by the
right-hand rule
:
- The
International Astronomical Union
(IAU) defines the
north pole
of a planet as that which lies on Earth's north side of the
invariable plane
of the
Solar System
;
[2]
under this system,
Venus
is tilted 3° and rotates
retrograde
, opposite that of most of the other planets.
[3]
[4]
- The IAU also uses the right-hand rule to define a
positive pole
[5]
for the purpose of determining orientation. Using this convention, Venus is tilted 177° ("upside down") and rotates prograde.
Earth
[
edit
]
Earth
's
orbital plane
is known as the
ecliptic
plane, and
Earth's tilt
is known to astronomers as the
obliquity of the ecliptic
, being the angle between the ecliptic and the
celestial equator
on the
celestial sphere
.
[6]
It is denoted by the
Greek letter
ε
.
Earth currently has an axial tilt of about 23.44°.
[7]
This value remains about the same relative to a stationary orbital plane throughout the cycles of
axial precession
.
[8]
But the ecliptic (i.e., Earth's orbit) moves due to planetary
perturbations
, and the obliquity of the ecliptic is not a fixed quantity. At present, it is decreasing at a rate of about
46.8″
[9]
per
century
(see details in
Short term
below)
.
History
[
edit
]
The ancient Greeks had good measurements of the obliquity since about 350 BCE, when
Pytheas
of Marseilles measured the shadow of a
gnomon
at the summer solstice.
[10]
About 830 CE, the Caliph
Al-Mamun
of Baghdad directed his astronomers to measure the obliquity, and the result was used in the Arab world for many years.
[11]
In 1437,
Ulugh Beg
determined the Earth's axial tilt as 23°30′17″ (23.5047°).
[12]
During the
Middle Ages
, it was widely believed that both precession and Earth's obliquity oscillated around a mean value, with a period of 672 years, an idea known as
trepidation
of the equinoxes. Perhaps the first to realize this was incorrect (during historic time) was
Ibn al-Shatir
in the fourteenth century
[13]
and the first to realize that the obliquity is decreasing at a relatively constant rate was
Fracastoro
in 1538.
[14]
The first accurate, modern, western observations of the obliquity were probably those of
Tycho Brahe
from
Denmark
, about 1584,
[15]
although observations by several others, including
al-Ma'mun
,
al-Tusi
,
[16]
Purbach
,
Regiomontanus
, and
Walther
, could have provided similar information.
Seasons
[
edit
]
Earth
's axis remains tilted in the same direction with reference to the background stars throughout a year (regardless of where it is in its
orbit
) due to the
gyroscope effect
. This means that one pole (and the associated
hemisphere of Earth
) will be directed away from the Sun at one side of the orbit, and half an orbit later (half a year later) this pole will be directed towards the Sun. This is the cause of Earth's
seasons
.
Summer
occurs in the
Northern hemisphere
when the north pole is directed toward the Sun. Variations in Earth's axial tilt can influence the seasons and is likely a factor in long-term
climatic change
(also see
Milankovitch cycles
)
.
Oscillation
[
edit
]
Short term
[
edit
]
The exact angular value of the obliquity is found by observation of the motions of Earth and
planets
over many years. Astronomers produce new
fundamental ephemerides
as the accuracy of
observation
improves and as the understanding of the
dynamics
increases, and from these ephemerides various astronomical values, including the obliquity, are derived.
Annual
almanacs
are published listing the derived values and methods of use. Until 1983, the
Astronomical Almanac
's angular value of the mean obliquity for any date was calculated based on the
work of Newcomb
, who analyzed positions of the planets until about 1895:
- ε
= 23°27′8.26″ ? 46.845″
T
? 0.0059″
T
2
+
0.001
81
″
T
3
where
ε
is the obliquity and
T
is
tropical centuries
from
B1900.0
to the date in question.
[17]
From 1984, the
Jet Propulsion Laboratory's DE series
of computer-generated ephemerides took over as the
fundamental ephemeris
of the
Astronomical Almanac
. Obliquity based on DE200, which analyzed observations from 1911 to 1979, was calculated:
- ε
= 23°26′21.448″ ? 46.8150″
T
? 0.00059″
T
2
+
0.001
813
″
T
3
where hereafter
T
is
Julian centuries
from
J2000.0
.
[18]
JPL's fundamental ephemerides have been continually updated. For instance, according to IAU resolution in 2006 in favor of the P03 astronomical model, the
Astronomical Almanac
for 2010 specifies:
[19]
- ε
= 23°26′21.406″ ?
46.836
769
″
T
?
0.000
1831
″
T
2
+
0.002
003
40
″
T
3
? 5.76″ × 10
?7
T
4
? 4.34″ × 10
?8
T
5
These expressions for the obliquity are intended for high precision over a relatively short time span, perhaps
±
several centuries.
[20]
Jacques Laskar
computed an expression to order
T
10
good to 0.02″ over 1000 years and several
arcseconds
over 10,000 years.
- ε
= 23°26′21.448″ ? 4680.93″
t
? 1.55″
t
2
+ 1999.25″
t
3
? 51.38″
t
4
? 249.67″
t
5
? 39.05″
t
6
+ 7.12″
t
7
+ 27.87″
t
8
+ 5.79″
t
9
+ 2.45″
t
10
where here
t
is multiples of 10,000
Julian years
from
J2000.0
.
[21]
These expressions are for the so-called
mean
obliquity, that is, the obliquity free from short-term variations. Periodic motions of the Moon and of Earth in its orbit cause much smaller (9.2
arcseconds
) short-period (about 18.6 years) oscillations of the rotation axis of Earth, known as
nutation
, which add a periodic component to Earth's obliquity.
[22]
[23]
The
true
or instantaneous obliquity includes this nutation.
[24]
Long term
[
edit
]
Using
numerical methods
to simulate
Solar System
behavior over a period of several million years, long-term changes in Earth's
orbit
, and hence its obliquity, have been investigated. For the past 5 million years, Earth's obliquity has varied between
22°2′33″
and
24°30′16″
, with a mean period of 41,040 years. This cycle is a combination of precession and the largest
term
in the motion of the
ecliptic
. For the next 1 million years, the cycle will carry the obliquity between
22°13′44″
and
24°20′50″
.
[25]
The
Moon
has a stabilizing effect on Earth's obliquity. Frequency map analysis conducted in 1993 suggested that, in the absence of the Moon, the obliquity could change rapidly due to
orbital resonances
and
chaotic behavior of the Solar System
, reaching as high as 90° in as little as a few million years (
also see
Orbit of the Moon
).
[26]
[27]
However, more recent numerical simulations
[28]
made in 2011 indicated that even in the absence of the Moon, Earth's obliquity might not be quite so unstable; varying only by about 20?25°. To resolve this contradiction, diffusion rate of obliquity has been calculated, and it was found that it takes more than billions of years for Earth's obliquity to reach near 90°.
[29]
The Moon's stabilizing effect will continue for less than two billion years. As the Moon continues to recede from Earth due to
tidal acceleration
, resonances may occur which will cause large oscillations of the obliquity.
[30]
Long-term obliquity of the ecliptic.
Left
: for the past 5 million years; note that the obliquity varies only from about 22.0° to 24.5°.
Right
: for the next 1 million years; note the approx. 41,000-year period of variation. In both graphs, the red point represents the year 1850.
(Source: Berger, 1976.)
Solar System bodies
[
edit
]
All four of the innermost, rocky planets of the
Solar System
may have had large variations of their obliquity in the past. Since obliquity is the angle between the axis of rotation and the direction perpendicular to the orbital plane, it changes as the orbital plane changes due to the influence of other planets. But the axis of rotation can also move (
axial precession
), due to torque exerted by the Sun on a planet's equatorial bulge. Like Earth, all of the rocky planets show axial precession. If the precession rate were very fast the obliquity would actually remain fairly constant even as the orbital plane changes.
[31]
The rate varies due to
tidal dissipation
and
core
-
mantle
interaction, among other things. When a planet's precession rate approaches certain values,
orbital resonances
may cause large changes in obliquity. The amplitude of the contribution having one of the resonant rates is divided by the difference between the resonant rate and the precession rate, so it becomes large when the two are similar.
[31]
Mercury
and
Venus
have most likely been stabilized by the tidal dissipation of the Sun. Earth was stabilized by the Moon, as mentioned above, but before its
formation
, Earth, too, could have passed through times of instability.
Mars
's obliquity is quite variable over millions of years and may be in a chaotic state; it varies as much as 0° to 60° over some millions of years, depending on
perturbations
of the planets.
[26]
[32]
Some authors dispute that Mars's obliquity is chaotic, and show that tidal dissipation and viscous core-mantle coupling are adequate for it to have reached a fully damped state, similar to Mercury and Venus.
[3]
[33]
The occasional shifts in the axial tilt of Mars have been suggested as an explanation for the appearance and disappearance of rivers and lakes over the course of the existence of Mars. A shift could cause a burst of methane into the atmosphere, causing warming, but then the methane would be destroyed and the climate would become arid again.
[34]
[35]
The obliquities of the outer planets are considered relatively stable.
Axis and rotation of selected Solar System bodies
Body
|
NASA
,
J2000
.0
[36]
epoch
|
IAU
, 0h 0 January 2010
TT
[37]
epoch
|
Axial tilt
(degrees)
|
North Pole
|
Rotational
period
(hours)
|
Axial tilt
(degrees)
|
North Pole
|
Rotation
(deg./day)
|
R.A.
(degrees)
|
Dec.
(degrees)
|
R.A.
(degrees)
|
Dec.
(degrees)
|
Sun
|
7.25
|
286.13
|
63.87
|
609.12
[A]
|
7.25
[B]
|
286.15
|
63.89
|
14.18
|
Mercury
|
0.03
|
281.01
|
61.41
|
1407.6
|
0.01
|
281.01
|
61.45
|
6.14
|
Venus
|
2.64
|
272.76
|
67.16
|
?5832.6
|
2.64
|
272.76
|
67.16
|
?1.48
|
Earth
|
23.44
|
0.00
|
90.00
|
23.93
|
23.44
|
Undefined
|
90.00
|
360.99
|
Moon
|
6.68
|
?
|
?
|
655.73
|
1.54
[C]
|
270.00
|
66.54
|
13.18
|
Mars
|
25.19
|
317.68
|
52.89
|
24.62
|
25.19
|
317.67
|
52.88
|
350.89
|
Jupiter
|
3.13
|
268.06
|
64.50
|
9.93
[D]
|
3.12
|
268.06
|
64.50
|
870.54
[D]
|
Saturn
|
26.73
|
40.59
|
83.54
|
10.66
[D]
|
26.73
|
40.59
|
83.54
|
810.79
[D]
|
Uranus
|
82.23
|
257.31
|
?15.18
|
?17.24
[D]
|
82.23
|
257.31
|
?15.18
|
?501.16
[D]
|
Neptune
|
28.32
|
299.33
|
42.95
|
16.11
[D]
|
28.33
|
299.40
|
42.95
|
536.31
[D]
|
Pluto
[E]
|
57.47
|
312.99
[E]
|
6.16
[E]
|
?153.29
|
60.41
|
312.99
|
6.16
|
?56.36
|
- ^
At 16° latitude; the Sun's rotation varies with latitude.
- ^
With respect to the
ecliptic
of 1850.
- ^
With respect to the ecliptic; the Moon's orbit is inclined 5.16° to the ecliptic.
- ^
a
b
c
d
e
f
g
h
From the origin of the radio emissions; the visible clouds generally rotate at different rate.
- ^
a
b
c
NASA lists the coordinates of Pluto's positive pole; noted values have been reinterpreted to correspond to the north/negative pole.
|
The stellar obliquity
ψ
s
, i.e. the axial tilt of a star with respect to the orbital plane of one of its planets, has been determined for only a few systems. By 2012, 49 stars have had sky-projected spin-orbit misalignment
λ
has been observed,
[38]
which serves as a lower limit to
ψ
s
. Most of these measurements rely on the
Rossiter?McLaughlin effect
. Since the launch of space-based telescopes such as
Kepler space telescope
, it has been made possible to determine and estimate the obliquity of an extrasolar planet. The rotational flattening of the planet and the entourage of moons and/or rings, which are traceable with high-precision photometry provide access to planetary obliquity,
ψ
p
. Many extrasolar planets have since had their obliquity determined, such as
Kepler-186f
and
Kepler-413b
.
[39]
[40]
Astrophysicists have applied tidal theories to predict the obliquity of
extrasolar planets
. It has been shown that the obliquities of exoplanets in the
habitable zone
around low-mass stars tend to be eroded in less than 10
9
years,
[41]
[42]
which means that they would not have tilt-induced seasons as Earth has.
See also
[
edit
]
References
[
edit
]
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PMID
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S2CID
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