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8b77098
constants: Add orbital period (Porb) for the nine planets
Wiljea Jun 1, 2026
a49cef7
constants: Compute planet T0 as live perihelion passage (IFTE)
Wiljea Jun 1, 2026
81dc74c
constants: Refine giant-planet Porb (anomalistic) and add T₀ uncertai…
Wiljea Jun 7, 2026
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203 changes: 160 additions & 43 deletions doc/calc-help/constants.md
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Expand Up @@ -1091,14 +1091,27 @@ It is the measured angle between the ascending node (where the orbit crosses
the reference plane northward) and the perihelion (the closest point to the
Sun), measured in the direction of motion. [Reference 23](#reference-23)

### Porb☿ constant

Mercury orbital period

Mercury's sidereal orbital period — the time to complete one revolution around
the Sun. It is distinct from the rotation period `Prot☿` (the spin period).
Computed from Meeus' Chapter 38 orbital elements and cross-checked against JPL
Horizons. The stored value is a mean orbital period, used as an approximation of
the anomalistic period that governs the recurrence of the perihelion passages
exposed by `T₀☿`. [Materials 22](#materials-22) [Reference 23](#reference-23)

### T₀☿ constant

Mercury last perihelion passage
Mercury time of perihelion passage

Mercury's most recent time of perihelion passage is a measured quantity.
Mercury passes perihelion approximately four times per year. Value in JDN
(Julian Day Number, Gregorian calendar). It is the most recent point in its
orbit when it was closest to the Sun. [Materials 20](#materials-20)
Computed, not stored: `T₀☿` evaluates an expression (IFTE) that advances from a
reference perihelion passage (Tp) by whole orbital periods (`Porb☿`) and returns
the most recent perihelion passage at or before the current date. The
computation is carried out in Julian Day Number and converted to a date with
`JDN→`, so the constant displays as a date but tracks "now". Mercury passes
perihelion roughly four times per year. [Materials 22](#materials-22)


## Venus constants
Expand Down Expand Up @@ -1205,13 +1218,26 @@ It is the measured angle between the ascending node (where the orbit crosses
the reference plane northward) and the perihelion (the closest point to the Sun),
measured in the direction of motion. [Reference 23](#reference-23)

### Porb♀ constant

Venus orbital period

Venus's sidereal orbital period — the time to complete one revolution around the
Sun. It is distinct from the rotation period `Prot♀` (the spin period). Computed
from Meeus' Chapter 38 orbital elements and cross-checked against JPL Horizons.
The stored value is a mean orbital period, used as an approximation of the
anomalistic period that governs the recurrence of the perihelion passages
exposed by `T₀♀`. [Materials 22](#materials-22) [Reference 23](#reference-23)

### T₀♀ constant

Venus last perihelion passage
Venus time of perihelion passage

Measured. Venus's most recent time of perihelion passage. Venus passes perihelion
approximately twice per year. Value in JDN. It is the most recent point in its
orbit when it was closest to the Sun. [Materials 20](#materials-20)
Computed, not stored: `T₀♀` evaluates an expression (IFTE) that advances from a
reference perihelion passage (Tp) by whole orbital periods (`Porb♀`) and returns
the most recent perihelion passage at or before the current date. The
computation is carried out in Julian Day Number and converted to a date with
`JDN→`. Venus passes perihelion about twice per year. [Materials 22](#materials-22)


## Earth constants
Expand Down Expand Up @@ -1324,13 +1350,25 @@ It is the measured angle between the ascending node (where the orbit crosses
the reference plane northward) and the perihelion (the closest point to the
Sun), measured in the direction of motion. [Reference 23](#reference-23)

### Porb♁ constant

Earth orbital period

Earth's sidereal orbital period — the time to complete one revolution around the
Sun. It is distinct from the rotation period `Prot♁` (the sidereal day). Computed
from Meeus' Chapter 38 orbital elements and cross-checked against JPL Horizons.
The stored value is the mean (anomalistic) year used to advance `T₀♁` between
successive perihelion passages. [Materials 22](#materials-22) [Reference 23](#reference-23)

### T₀♁ constant

Earth perihelion passage
Earth time of perihelion passage

Measured. Earth's most recent time of perihelion passage (early January each
year). Value in JDN. It is the most recent point in its orbit when it was
closest to the Sun. [Materials 20](#materials-20)
Computed, not stored: `T₀♁` evaluates an expression (IFTE) that advances from a
reference perihelion passage (Tp) by whole orbital periods (`Porb♁`, the
anomalistic year) and returns the most recent perihelion passage at or before
the current date (early January each year). The computation is carried out in
Julian Day Number and converted to a date with `JDN→`. [Materials 22](#materials-22)


## Moon constants
Expand Down Expand Up @@ -1552,13 +1590,26 @@ It is the measured angle between the ascending node (where the orbit crosses
the reference plane northward) and the perihelion (the closest point to the
Sun), measured in the direction of motion. [Reference 23](#reference-23)

### Porb♂ constant

Mars orbital period

Mars's sidereal orbital period — the time to complete one revolution around the
Sun. It is distinct from the rotation period `Prot♂` (the spin period, close to
an Earth day). Computed from Meeus' Chapter 38 orbital elements and cross-checked
against JPL Horizons. The stored value is a mean orbital period, used as an
approximation of the anomalistic period that governs the recurrence of the
perihelion passages exposed by `T₀♂`. [Materials 22](#materials-22) [Reference 23](#reference-23)

### T₀♂ constant

Mars perihelion passage
Mars time of perihelion passage

Measured. Mars's most recent time of perihelion passage. Mars's orbital
period is approximately 1.88 years. Value in JDN. It is the most recent point
in its orbit when it was closest to the Sun. [Materials 20](#materials-20)
Computed, not stored: `T₀♂` evaluates an expression (IFTE) that advances from a
reference perihelion passage (Tp) by whole orbital periods (`Porb♂`) and returns
the most recent perihelion passage at or before the current date. The
computation is carried out in Julian Day Number and converted to a date with
`JDN→`. Mars's orbital period is approximately 1.88 years. [Materials 22](#materials-22)


## Jupiter constants
Expand Down Expand Up @@ -1670,13 +1721,25 @@ It is the measured angle between the ascending node (where the orbit
crosses the reference plane northward) and the perihelion (the closest
point to the Sun), measured in the direction of motion. [Reference 23](#reference-23)

### Porb♃ constant

Jupiter orbital period

Jupiter's orbital period — the time to complete one revolution around the Sun
(about 11.86 years). It is distinct from the rotation period `Prot♃` (the ~10 h
spin period). The value is the anomalistic period (perihelion to perihelion),
computed from Meeus' Chapter 38, which governs the recurrence of the perihelion
passages exposed by `T₀♃`. [Materials 22](#materials-22)

### T₀♃ constant

Jupiter perihelion passage
Jupiter time of perihelion passage

Measured. Jupiter's most recent time of perihelion passage. Jupiter's
orbital period is approximately 11.86 years. It is the most recent point
in its orbit when it was closest to the Sun. Value in JDN. [Reference 4](#reference-4) [Materials 21](#materials-21)
Computed, not stored: `T₀♃` evaluates an expression (IFTE) that advances from a
reference perihelion passage (Tp) by whole orbital periods (`Porb♃`) and returns
the most recent perihelion passage at or before the current date. The
computation is carried out in Julian Day Number and converted to a date with
`JDN→`. Jupiter's orbital period is approximately 11.86 years. [Materials 22](#materials-22)


## Saturn constants
Expand Down Expand Up @@ -1787,13 +1850,25 @@ It is the measured angle between the ascending node (where the orbit
crosses the reference plane northward) and the perihelion (the closest
point to the Sun), measured in the direction of motion. [Reference 23](#reference-23)

### Porb♄ constant

Saturn orbital period

Saturn's orbital period — the time to complete one revolution around the Sun
(about 29.46 years). It is distinct from the rotation period `Prot♄` (the ~10.7 h
spin period). The value is the anomalistic period (perihelion to perihelion),
computed from Meeus' Chapter 38, which governs the recurrence of the perihelion
passages exposed by `T₀♄`. [Materials 22](#materials-22)

### T₀♄ constant

Saturn perihelion passage
Saturn time of perihelion passage

Measured. Saturn's most recent time of perihelion passage. Saturn's
orbital period is approximately 29.46 years. It is the most recent point
in its orbit when it was closest to the Sun. Value in JDN. [Reference 4](#reference-4) [Materials 21](#materials-21)
Computed, not stored: `T₀♄` evaluates an expression (IFTE) that advances from a
reference perihelion passage (Tp) by whole orbital periods (`Porb♄`) and returns
the most recent perihelion passage at or before the current date. The
computation is carried out in Julian Day Number and converted to a date with
`JDN→`. Saturn's orbital period is approximately 29.46 years. [Materials 22](#materials-22)


## Uranus constants
Expand Down Expand Up @@ -1903,15 +1978,27 @@ It is the measured angle between the ascending node (where the orbit
crosses the reference plane northward) and the perihelion (the closest
point to the Sun), measured in the direction of motion. [Reference 23](#reference-23)

### Porb⛢ constant

Uranus orbital period

Uranus's orbital period — the time to complete one revolution around the Sun
(about 84 years). It is distinct from the rotation period `Prot⛢` (the ~17 h spin
period). The value is the anomalistic period (perihelion to perihelion), computed
from Meeus' Chapter 38, which governs the recurrence of the perihelion passages
exposed by `T₀⛢`. [Materials 22](#materials-22)

### T₀⛢ constant

Uranus perihelion passage
Uranus time of perihelion passage

Calculated from measurement (JPL DE440 orbital propagation). Uranus's
predicted next time of perihelion passage (~2050). Uranus's orbital
period is approximately 84 years. Last perihelion: 1966. It is the most
recent point in its orbit when it was closest to the Sun. Value in JDN.
[Reference 4](#reference-4) [Materials 21](#materials-21)
Computed, not stored: `T₀⛢` evaluates an expression (IFTE) that advances from a
reference perihelion passage (Tp) by whole orbital periods (`Porb⛢`) and returns
the most recent perihelion passage at or before the current date. The
computation is carried out in Julian Day Number and converted to a date with
`JDN→`. Uranus's orbital period is approximately 84 years; the last perihelion
was in 1966 and the next is around 2050, so the returned value can be decades in
the past. [Materials 22](#materials-22)


## Neptune constants
Expand Down Expand Up @@ -2020,15 +2107,28 @@ It is the measured angle between the ascending node (where the orbit
crosses the reference plane northward) and the perihelion (the closest
point to the Sun), measured in the direction of motion. [Reference 23](#reference-23)

### Porb♆ constant

Neptune orbital period

Neptune's sidereal orbital period — the time to complete one revolution around
the Sun (about 165 years). It is distinct from the rotation period `Prot♆` (the
~16 h spin period). Tabulated sidereal value from the NASA Planetary Fact Sheet,
used as an approximation of the anomalistic period that governs the recurrence
of the perihelion passages exposed by `T₀♆`. [Materials 23](#materials-23)

### T₀♆ constant

Neptune perihelion passage
Neptune time of perihelion passage

Calculated from measurement (JPL DE440 orbital propagation). Neptune's
predicted next time of perihelion passage (~2042). Neptune's orbital
period is approximately 164.8 years. Last perihelion: 1876. It is the
most recent point in its orbit when it was closest to the Sun. Value
in JDN. [Reference 4](#reference-4) [Materials 21](#materials-21)
Computed, not stored: `T₀♆` evaluates an expression (IFTE) that advances from a
reference perihelion passage (Tp) by whole orbital periods (`Porb♆`) and returns
the most recent perihelion passage at or before the current date, carried out in
Julian Day Number and converted to a date with `JDN→`. Neptune is a deliberate
exception to the "most recent past perihelion" convention: its true last
perihelion (~1876) is uninformative and hard to source, so Tp is set to the next
perihelion (2042-09-04). Because that date is in the future, the floor-IFTE
returns it unchanged, giving the useful upcoming date. [Materials 21](#materials-21)


## Pluto constants
Expand Down Expand Up @@ -2135,14 +2235,27 @@ It is the measured angle between the ascending node (where the orbit crosses
the reference plane northward) and the perihelion (the closest point to the
Sun), measured in the direction of motion. [Reference 23](#reference-23)

### Porb♇ constant

Pluto orbital period

Pluto's sidereal orbital period — the time to complete one revolution around the
Sun (about 248 years). It is distinct from the rotation period `Prot♇` (the
~6.4 day spin period). Tabulated sidereal value from the NASA Planetary Fact
Sheet, used as an approximation of the anomalistic period that governs the
recurrence of the perihelion passages exposed by `T₀♇`. [Materials 23](#materials-23)

### T₀♇ constant

Pluto perihelion passage
Pluto time of perihelion passage

Measured. Pluto's last time of perihelion passage (1989 Sep 05). Pluto's
orbital period is approximately 248 years. It is the most recent point in
its orbit when it was closest to the Sun. Next perihelion: ~2237. Value
in JDN. [Reference 4](#reference-4) [Materials 21](#materials-21)
Computed, not stored: `T₀♇` evaluates an expression (IFTE) that advances from a
reference perihelion passage (Tp) by whole orbital periods (`Porb♇`) and returns
the most recent perihelion passage at or before the current date. The
computation is carried out in Julian Day Number and converted to a date with
`JDN→`. Pluto's orbital period is approximately 248 years; the last perihelion
was in 1989 (Sep 05) and the next is around 2237, so the returned value can be
decades in the past. [Materials 21](#materials-21)

## Sun constants

Expand Down Expand Up @@ -3506,4 +3619,8 @@ Park, R.S., et al. (2021). "The JPL Planetary and Lunar Ephemerides DE440 and DE

### Materials 22

Meeus, J. (1998). Astronomical Algorithms, 2nd ed. Willmann-Bell Inc., Richmond, Virginia. ISBN: 978-0-943396-61-3. Chapter 50: Perigee and Apogee of the Moon, pp. 355–358. (Formula for lunar perigee — implemented in MPERC.txt.)
Meeus, J. (1998). Astronomical Algorithms, 2nd ed. Willmann-Bell Inc., Richmond, Virginia. ISBN: 978-0-943396-61-3. Chapter 38: Perihelion and Aphelion of the Planets; Chapter 50: Perigee and Apogee of the Moon, pp. 355–358. (Planetary orbital periods derived from the Chapter 38 elements; lunar perigee formula from Chapter 50.)

### Materials 23

Williams, D.R. NASA Planetary Fact Sheet. NASA Space Science Data Coordinated Archive (NSSDCA), Goddard Space Flight Center. [Source](https://nssdc.gsfc.nasa.gov/planetary/factsheet/) Accessed: June 2026. (Sidereal orbital periods of the giant planets, used as an approximation of the anomalistic period.)
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