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Synodic day
Synodic day
Synodic day
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A synodic day (or synodic rotation period or solar day) is the period for a celestial object to rotate once in relation to the star it is orbiting, and is the basis of solar time.

The synodic day is distinguished from the sidereal day, which is one complete rotation in relation to distant stars[1] and is the basis of sidereal time.

In the case of a tidally locked planet, the same side always faces its parent star, and its synodic day is infinite. Its sidereal day, however, is equal to its orbital period.

Earth

[edit]
Main articles: Earth's rotation, Solar time, and Day

Earth's synodic day is the time it takes for the Sun to pass over the same meridian (a line of longitude) on consecutive days, whereas a sidereal day is the time it takes for a given distant star to pass over a meridian on consecutive days.[2] For example, in the Northern Hemisphere, a synodic day could be measured as the time taken for the Sun to move from exactly true south (i.e. its highest declination) on one day to exactly south again on the next day (or exactly true north in the Southern Hemisphere).

Derivative of −Δt. The axis on the right shows the length of the solar day.

For Earth, the synodic day is not constant, and changes over the course of the year due to the eccentricity of Earth's orbit around the Sun and the axial tilt of the Earth.[3] The longest and shortest synodic days' durations differ by about 51 seconds.[4] The mean length, however, is 24 hours (with fluctuations on the order of milliseconds), and is the basis of solar time. The difference between the mean and apparent solar time is the equation of time, which can also be seen in Earth's analemma. Because of the variation in the length of the synodic day, the days with the longest and shortest period of daylight do not coincide with the solstices near the equator.

As viewed from Earth during the year, the Sun appears to slowly drift along an imaginary path coplanar with Earth's orbit, known as the ecliptic, on a spherical background of seemingly fixed stars.[5] Each synodic day, this gradual motion is a little less than 1° eastward (360° per 365.25 days), in a manner known as prograde motion.

Certain spacecraft orbits, Sun-synchronous orbits, have orbital periods that are a fraction of a synodic day. Combined with a nodal precession, this allows them to always pass over a location on Earth's surface at the same mean solar time.[6]

Moon

[edit]
Main article: Lunar day

Due to tidal locking with Earth, the Moon's synodic day (the lunar day or synodic rotation period) is the same as its synodic period with Earth and the Sun (the period of the lunar phases, the synodic lunar month, which is the month of the lunar calendar).

Venus

[edit]
Further information: Venus § Orbit and rotation

Due to the slow retrograde rotational speed of Venus, its synodic rotation period of 117 Earth days is about half the length of its sidereal rotational period (sidereal day) and even its orbital period.[7]

Mercury

[edit]
Further information: Mercury (planet) § Spin-orbit resonance

Due to Mercury's slow rotational speed and fast orbit around the Sun, its synodic rotation period of 176 Earth days is three times longer than its sidereal rotational period (sidereal day) and twice as long as its orbital period.[8]

See also

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References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Synodic day

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from Grokipedia
The synodic day, also known as the solar day, is the period for a celestial body to complete one rotation relative to its parent star, such as the time for the Sun to return to the same position in the sky relative to a fixed point on the body's surface. For , this interval averages 24 hours. This duration represents the mean solar day, which accounts for slight variations in the Sun's apparent motion caused by 's elliptical orbit and . In contrast to the sidereal day, which measures Earth's rotation relative to distant stars and lasts approximately 23 hours, 56 minutes, and 4 seconds, the synodic day is about longer due to Earth's orbital motion . During one sidereal day, Earth completes a full 360-degree on its axis, but for the synodic day, it must rotate slightly more—about 361 degrees—to compensate for the roughly 1-degree eastward shift in the Sun's position caused by the planet's daily orbital progress. This difference results in there being 365 mean solar days in a year but 366 sidereal days. The synodic day forms the basis for civil timekeeping worldwide, defining the standard 24-hour day used in calendars and daily life, while the sidereal day is primarily employed in astronomy for precise observations of celestial objects. Apparent solar days vary slightly throughout the year—ranging from about 20 seconds shorter to 30 seconds longer than 24 hours—due to the combined effects of Earth's and obliquity, leading to the equation of time and the figure-eight pattern traced by the Sun. Understanding the synodic day is essential for distinguishing between apparent and mean in horology and .

Fundamentals

Definition

The synodic day, also known as the solar day, is the duration of one complete rotation of a celestial body relative to the Sun, measured from one solar noon—when the Sun reaches its highest point in the sky, crossing the local meridian—to the next such occurrence. This interval accounts for both the body's axial rotation and its orbital motion around the Sun, making it the basis for on that body. The concept of the synodic day originated from ancient observations of daily solar cycles, where early astronomers tracked the Sun's apparent path across the to establish fundamental timekeeping. The term "synodic" derives from the Greek "synodos," meaning conjunction or meeting, which in this context refers to the periodic alignment of the celestial body with the Sun. Synodic days are typically expressed in units of hours or equivalent days, scaled to the specific celestial body's rotational characteristics. Observational basis lies in the apparent motion of the Sun as seen from the body's surface, driven by the interplay of rotation and orbital progression.

Difference from Sidereal Day

The sidereal day represents the time required for a celestial body to complete one full on its axis relative to the , serving as a direct measure of its rotational period. In contrast, the synodic day measures the interval between successive transits of the Sun across the body's meridian, incorporating both its rotation and its orbital motion around the Sun. This fundamental difference arises because the synodic day must account for the body's orbital advance during each rotation; for prograde-rotating bodies orbiting the Sun, the (or analogous body) moves approximately 1 degree along its orbit per day, requiring an additional rotation to realign the Sun's position, thus making the synodic day longer than the sidereal day by that orbital increment. On , for instance, this results in a sidereal day that is about 4 minutes shorter than the synodic (solar) day, reflecting the annual orbital progress of roughly 1 degree per day. Practically, the synodic day governs daylight cycles, the structure of calendars, and everyday human timekeeping systems, as it aligns with observable solar positions essential for agriculture, navigation, and societal routines. The sidereal day, however, is primarily utilized in astronomy for precise tracking of celestial objects, enabling accurate observations without the variability introduced by orbital motion. For bodies with retrograde rotation, such as , the synodic day can be shorter in duration than the sidereal day, as the opposing directions of and orbital motion partially counteract each other in the solar alignment.

Mathematical Formulation

General Formula

The length of the synodic day, denoted as SS, for a celestial body orbiting the Sun is given by the formula S=11R−1Y,S = \frac{1}{\frac{1}{R} - \frac{1}{Y}}, where RR is the sidereal period of the body and YY is its sidereal around the Sun, both expressed in consistent units such as days; this assumes prograde rotation and R<YR < Y. This formula arises from the relative angular velocity of the body's rotation with respect to the Sun. The sidereal rotation angular velocity is ωr=2π/R\omega_r = 2\pi / R, and the orbital angular velocity is ωo=2π/Y\omega_o = 2\pi / Y. The effective angular velocity relative to the Sun is then ωs=ωr−ωo\omega_s = \omega_r - \omega_o, yielding the synodic day length as S=2π/ωsS = 2\pi / \omega_s, which simplifies to the expression above upon substitution. The formula applies to bodies orbiting the Sun under first-order approximations that neglect orbital eccentricity and axial obliquity, assuming a circular orbit for simplicity. For bodies with retrograde rotation, the sidereal rotation period RR is conventionally taken as negative to account for the opposite direction relative to the orbital motion. This adjusts the formula to S=11R+1YS = \frac{1}{\frac{1}{R} + \frac{1}{Y}} (with ∣R∣|R| used in the denominator), as the orbital motion effectively adds to the rotation rate in the relative frame, resulting in a positive synodic day length but in the retrograde direction.

Derivation and Factors

The synodic day arises from the relative motion between a planet's rotation and its orbital revolution around the central star. To derive its length from first principles, consider the angular rotation rate ωr=360∘/R\omega_r = 360^\circ / R, where RR is the sidereal rotation period in days. The orbital angular rate is ωo=360∘/Y\omega_o = 360^\circ / Y, where YY is the sidereal orbital period in days. For prograde rotation (the typical case in the solar system), the effective angular rate relative to the Sun is ωs=ωr−ωo\omega_s = \omega_r - \omega_o, as the orbital motion carries the planet eastward, reducing the apparent speed of the Sun across the sky. The synodic day length SS is then the time for the Sun to complete one full apparent cycle: S=360∘/ωs=1/(1/R−1/Y)S = 360^\circ / \omega_s = 1 / (1/R - 1/Y). This simplifies to S=RY/(Y−R)S = RY / (Y - R), assuming Y>RY > R. This derivation assumes a and constant rates, but several astronomical factors modify the synodic day length. Orbital introduces variability because the planet's orbital speed changes according to Kepler's second law: slower near aphelion (farthest from the Sun) and faster near perihelion. This causes the synodic day to lengthen when orbital motion is slower, as the relative angular rate ωs\omega_s decreases; for , this contributes to apparent solar day variations of up to about 20 seconds over the year. , or obliquity, influences the duration of daylight hours seasonally by altering the Sun's path across the sky but does not affect the mean synodic day length, which is defined as the interval between successive solar transits and remains governed by the rotational and orbital periods. Spin-orbit resonances further alter the effective synodic day by coupling the and orbital periods through tidal interactions. In such cases, the rotation rate locks to a rational multiple of the orbital rate, overriding the standard derivation; for example, Mercury's 3:2 means it rotates three times for every two orbits, yielding a synodic day of approximately 176 days (precisely 175.95 days), as the Sun returns to the meridian after two full orbits. Approximations simplify the formula for extreme cases: for fast rotators like gas giants, where R≪YR \ll Y (hours versus decades), ωo\omega_o is negligible, so S≈RS \approx R; for slow rotators like (retrograde, with R>YR > Y), the formula adjusts to S=1/(1/R+1/Y)S = 1 / (1/R + 1/Y) due to the opposing directions, making orbital motion dominant and resulting in a synodic day much longer than the sidereal . Relativistic effects, such as general relativity's influence on orbital and , are negligible for synodic day lengths across solar system bodies, introducing perturbations on the order of milliarcseconds in rotation parameters that do not measurably alter the periods; minor adjustments occur near the Sun for Mercury, but these are far smaller than classical factors.

Solar System Bodies

Earth

The synodic day on , also referred to as the solar day, serves as the fundamental unit for civil timekeeping and is defined as the interval between successive meridian transits of the Sun. By international agreement since 1967, when the second was redefined in terms of cesium-133 atomic transitions, the civil day used in timekeeping is exactly 24 hours, or 86,400 SI seconds, providing a uniform standard despite natural variations in and the slightly longer astronomical solar day. This definition underpins global time systems, ensuring consistency for , , and daily life. In contrast to the sidereal day, which measures relative to distant and lasts approximately 23 hours, 56 minutes, and 4 seconds, the synodic day is longer due to Earth's orbital motion . Over the course of a year, comprising about 365.25 days, this orbital progression adds roughly 4 minutes to each rotation cycle, aligning the Sun's apparent position after a full 360-degree turn plus the daily advance in orbit. The apparent solar day, based on the actual Sun's position, exhibits short-term fluctuations of about ±20 to 30 seconds from the mean, influenced by Earth's elliptical orbit and axial obliquity through the equation of time, as well as minor tidal effects. The equation of time quantifies the cumulative discrepancy between apparent and mean , reaching extremes of approximately ±16 minutes annually—most positive in and most negative in —arising from the combined effects of and the 23.44-degree tilt of Earth's axis. These variations mean that sundials, which track apparent solar time, can deviate significantly from clock time, necessitating adjustments in precise applications like astronomy. Historically, time measurement evolved from ancient sundials and water clocks, which approximated solar days, to mechanical clocks in the , and ultimately to the atomic standards of the ; civil time worldwide standardized on the mean solar day referenced to the Greenwich meridian following the 1884 . Over geological timescales, Earth's synodic day is gradually lengthening due to tidal friction from gravitational interactions with the , which dissipates through ocean bulges dragged ahead of the Moon's position. This secular trend amounts to an increase of about 1.7 milliseconds per century in the length of the day, a rate confirmed by analyses of ancient records and growth patterns, potentially extending future days beyond 24 hours in millions of years.

Moon

The synodic day on the , measured from one sunrise to the next at a given location, spans approximately 29.53059 days, corresponding directly to the length of one synodic month. This extended period arises from the 's with , where its sidereal rotation period equals its sidereal of 27.32166 days, keeping the same hemisphere perpetually facing . Relative to the Sun, however, the advances by one full orbit during each synodic month, synchronizing its solar day with this phase cycle. At the lunar surface, this configuration produces a day-night cycle of roughly 14.77 Earth days of continuous sunlight followed by an equal duration of darkness, excluding polar regions where illumination patterns differ. Such prolonged exposure to solar radiation and subsequent absence drives extreme temperature swings, reaching up to 127°C during daylight and dropping to -173°C at night, influencing surface processes like and regolith behavior. The Moon's synodic day underpins traditional lunar calendars, which divide the year into months aligned with the 29.5-day phase progression from new moon to new moon. Human exploration during the Apollo missions capitalized on this cycle by scheduling landings shortly after local sunrise, providing about 14 days of usable daylight; surface stays lasted up to three Earth days, enabling extensive extravehicular activities under consistent illumination. Although slight —oscillations in the Moon's orientation—allows up to 59% of its surface to be visible from over time and introduces minor positional variations, the mean synodic day length remains stable without significant alteration.

Venus

Venus's synodic day, or solar day—the time from one solar noon to the next—lasts 116.75 days, significantly shorter than its sidereal rotation period of 243.0226 days. This counterintuitive brevity arises from Venus's retrograde , in which the planet spins in the direction opposite to its orbital motion . As a result, the orbital motion effectively adds to the rotational speed relative to the fixed stars, accelerating the apparent motion of the Sun across the sky and thus shortening the solar day compared to the sidereal period. The planet's , or solar year, is 224.701 days, making the synodic day roughly half a Venusian year. Due to the retrograde spin, the Sun rises in the west and sets in the east, completing its cycle over this extended period. This configuration means that over one full , the Sun would trace out approximately 1.92 paths across the sky from an observer's perspective. Venus's thick, global of droplets, extending from about 48 to 70 km altitude, results in nearly uniform insolation across the planet's surface, with minimal variation in solar heating by or time of day. However, the prolonged duration of the synodic day contributes to extreme surface temperatures, averaging around 462°C (735 K), as the slow allows for significant accumulation under the intense of the carbon dioxide-dominated atmosphere. Unlike Mercury's locked 3:2 spin-orbit , Venus's rotation is not fully synchronized with its , representing a near-miss of such a configuration. The slow retrograde spin is likely the outcome of long-term tidal evolution, potentially influenced by atmospheric and gravitational interactions, or an ancient giant impact that reversed and decelerated the original prograde rotation.

Mercury

Mercury's synodic day, the time from one solar noon to the next as observed from its surface, lasts 175.94 days, equivalent to two Mercury years. This prolonged duration arises from the planet's 3:2 spin-orbit , in which Mercury completes exactly three s on its axis for every two orbits . The sidereal period is 58.646 days, while the sidereal is 87.969 days, meaning the planet's rotation rate slightly exceeds half its orbital rate, resulting in the Sun taking much longer to return to the same position in the sky. This is stabilized by solar tidal torques and friction, which capture and maintain Mercury's spin in this configuration despite its eccentric orbit (eccentricity of 0.2056). Near perihelion, the planet's increases such that the Sun's apparent motion across the slows, halts, and briefly reverses direction for observers on the surface, an effect spanning about eight hours and covering roughly one solar diameter eastward before resuming normal progression. This retrograde phenomenon, unique to Mercury's inner position and , contributes to its distinctive solar visibility patterns, akin to prolonged "morning star" phases during certain orbital points. The close proximity to the Sun and the extended synodic day lead to extreme surface variations, with daytime highs reaching 430°C (800°F) and nighttime lows dropping to -180°C (-290°F). Mercury's thin provides negligible insulation, amplifying these contrasts as heat rapidly dissipates after sunset, while intense solar radiation dominates the prolonged daylight.

Mars and Outer Planets

For Mars, the synodic day, known as a , lasts 24 hours, 39 minutes, and 35 seconds, while the sidereal rotation period is 24 hours, 37 minutes, and 23 seconds. This slight lengthening of the synodic day compared to the sidereal period arises from Mars's orbital motion around the Sun, with its sidereal of approximately 687 Earth days causing a modest discrepancy similar to 's. Mars thus serves as a transitional case between the inner planets and the outer giants, where rotational and orbital dynamics produce a noticeable but small difference. In contrast, for the outer planets—, Saturn, , and —their extremely long s relative to their rapid rotation rates result in synodic days that are nearly identical to sidereal rotation periods. This approximation holds because the orbital angular displacement per rotation is negligible when the orbital period YY greatly exceeds the sidereal rotation period RR, yielding S≈RS \approx R. The following table summarizes these periods:
PlanetSidereal Rotation PeriodSynodic Day (Approximate)Sidereal Orbital Period (Earth Years)
9 hours, 55 minutes, 30 seconds~9 hours, 55 minutes, 30 seconds11.86
Saturn10 hours, 33 minutes~10 hours, 33 minutes29.45
17 hours, 14 minutes (retrograde)~17 hours, 14 minutes (retrograde)84.02
16 hours, 6 minutes~16 hours, 6 minutes164.8
Jupiter's swift dominates its dynamics, with the negligible orbital influence over its 11.86-year path making the solar day effectively equal to the sidereal period. Saturn exhibits a similar , its 29.45-year contributing virtually no extension to the day length. Uranus's retrograde further underscores this trend, as its 84-year has minimal impact on the synodic period. Finally, Neptune's , informed by observations and subsequent analyses, aligns closely with its solar day amid a 164.8-year orbital cycle.

References

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