Conjunctions Examples

This notebook shows how MontuPython finds angular conjunctions between planets, stars, and mixed groups.

``Conjunction``: evaluate separation, visibility, and lapse at one epoch.
``ConjunctionExplorer``: scan a date interval for local minima below a threshold.

MontuPython uses the maximum pairwise separation among all body pairs. A conjunction is in range when that value is at or below maxseparation (default 5°).

If running in Google Colab, MontuPython must be installed first. In a local copy of the repository this cell can remain commented out.

[1]:

try:
    from google.colab import drive
    %pip install -Uq montu
except ImportError:
    print("Not running in Colab, skipping installation")
    import plotly.io as pio
    pio.renderers.default = "notebook_connected"
    %load_ext autoreload
    %autoreload 2
# Create folders for figures and temporal files
!mkdir -p ./gallery/ ./montu_dem/

Not running in Colab, skipping installation

[2]:

%matplotlib inline
import montu
import pandas as pd

pd.options.display.float_format = "{:.3f}".format

MontuPython version 0.50.0. 𓇍𓇋𓇋𓏏𓅓𓊵 𓎛𓎡𓄿𓀭𓎛𓈖𓂝𓎡 (ii-ti m Htp, HkAx Hn'-k)

Setup and conventions

``separation``: maximum pairwise angular separation [deg] at the epoch;
``in_conjunction``: True when separation <= maxseparation;
``visible_from_site``: all bodies above the horizon and the Sun below −5° (or n/a for geocentric runs);
``explore_lapse()``: UTC interval during which the group stays within maxseparation around the reference day.

Use return_as='Star' when selecting one star from montu.Stars.

[3]:

mars = montu.Planet('Mars')
aldebaran = montu.Stars(subset='bright', ProperName='Aldebaran', return_as='Star')

# Observers used throughout the notebook
medellin = montu.Observer(lat=6, lon=-75)
thebes = montu.Observer(site='thebes')
athens = montu.Observer(site='athens')

sites = {
    'geocentric': 'geocentric',
    'Medellín': medellin,
    'Thebes': thebes,
    'Athens': athens,
}

Loading stellar catalogue montu_stellar_catalogue_v38_bright.csv

ConjunctionExplorer — scan an interval

ConjunctionExplorer.search(start, end, observer=...) returns a list of fully computed Conjunction objects, one per qualifying local minimum.

[4]:

explorer = montu.ConjunctionExplorer(bodies=[mars, aldebaran], maxseparation=5)
conjs = explorer.search(
    start=montu.Time('2022-09-01'),
    end=montu.Time('2022-10-01'),
    observer='geocentric',
    verbose=True
)
for conj in conjs:
    conj.show_details()

100%|█████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 31/31 [00:00<00:00, 20274.98it/s]

Conjunction: Mars–Aldebaran
  Epoch (UTC)          : 2022-09-07 14:28:28
  Julian Day (UTC)     : 2459830.103152
  Observer             : geocentric
  Angular separation   : 4.2746° (max allowed 5.0°)
  In conjunction       : yes
  Is visible from site : n/a (geocentric)
  Pair Mars–Aldebaran
    Separation         : 4.2746°
    Position angle     : 170.19° (N→E)
  Mars
    Phase              : 85.42%
    Angular size       : 0.169 arcmin
    V magnitude        : -0.22
  Aldebaran
    V magnitude        : 0.87

Once a conjunction in a geocentric location, we can evaluate the condition at different locations

[5]:

conjunction = montu.Conjunction(
    bodies=[mars, aldebaran],
    maxseparation=5,
    mtime=conjs[0].mtime,
    observer=medellin,
)
conjunction.show_details()

Conjunction: Mars–Aldebaran
  Epoch (UTC)          : 2022-09-07 14:28:28
  Julian Day (UTC)     : 2459830.103152
  Observer             : lat 6.000000°, lon -75.000000°
  Local solar time     : 09:28:32.298
  Angular separation   : 4.2756° (max allowed 5.0°)
  In conjunction       : yes
  Sun altitude         : 52.85°
  Is visible from site : no (bodies above horizon and Sun < -5°)
  Pair Mars–Aldebaran
    Separation         : 4.2756°
    Position angle     : 170.18° (N→E)
  Mars
    Elevation / azimuth: 29.94° / 290.54° (above horizon: yes)
    Rise (UTC)         : 2022-09-08 04:14:14
    Set (UTC)          : 2022-09-07 16:38:38
    Phase              : 85.42%
    Angular size       : 0.169 arcmin
    V magnitude        : -0.22
  Aldebaran
    Elevation / azimuth: 30.95° / 285.72° (above horizon: yes)
    Rise (UTC)         : 2022-09-08 04:18:18
    Set (UTC)          : 2022-09-07 16:39:39
    V magnitude        : 0.87

As you can see the conjunction at minimum can’t be seen from this site. We can then explore times to check when is visible:

[6]:

conjunction = montu.Conjunction(
    bodies=[mars, aldebaran],
    maxseparation=5,
    mtime=conjs[0].mtime,
    observer=medellin,
)
lapse = conjunction.explore_lapse(verbose=False)
conjunction.plot_lapse(lapse[0], lapse[1], step_hours=1)

As you can see there are several dates and times at which the conjunction will be visible. For instance on 2022-09-07, 02:00 local time. Let’s see the conditions:

[7]:

lapse

[7]:

(Time('2022-09-02 11:59:18.095983'/'2022-09-02 11:59:59'/'[hrw 4806] I peret 18'/JED 2459824.999515/JTD 2459825.0003611),
 Time('2022-09-12 18:33:42.295690'/'2022-09-12 18:33:33'/'[hrw 4806] I peret 28'/JED 2459835.2734062/JTD 2459835.2742523))

[8]:

conditions = conjunction.is_visible(
    from_site=medellin,
    at=montu.Time('2022-09-07 02:00:00', zone=medellin),
    verbose=False,
)
montu.Util.print_dict(conditions)

| Key               | Value                               |
|-------------------|-------------------------------------|
| mtime             | 2022-09-07 07:00:00                 |
| observer          | lat 6.000000°, lon -75.000000°, 0 m |
| from_site         | lat 6.000000°, lon -75.000000°, 0 m |
| is_geocentric     | no                                  |
| separation        | 4.28                                |
| maxseparation     | 5.00                                |
| in_conjunction    | yes                                 |
| above_horizon     | yes                                 |
| sun_altitude      | -57.29                              |
| visible_from_site | yes                                 |
| visible           | yes                                 |
| body_conditions   | [2 rows — see below]                |
| pairs             | [1 rows — see below]                |

body_conditions:
| name      |    az |    el | above_horizon   |   ra_epoch |   dec_epoch |   vmag | rise_time           |   rise_az | set_time            |   set_az | phase   | angsize_arcmin   |   elongation |
|-----------|-------|-------|-----------------|------------|-------------|--------|---------------------|-----------|---------------------|----------|---------|------------------|--------------|
| Mars      | 68.39 | 37.54 | yes             |    4.55886 |      20.741 |  -0.22 | 2022-09-08 04:14:14 |     68.99 | 2022-09-07 16:38:38 |   290.96 | 85.40   | 0.17             |       -94.82 |
| Aldebaran | 73.75 | 37.18 | yes             |    4.62022 |      16.555 |   0.87 | 2022-09-08 04:18:18 |     73.29 | 2022-09-07 16:39:39 |   286.71 | —       | —                |       -94.54 |

pairs:
| bodies          |   separation_deg |   position_angle_deg | distance_au   |
|-----------------|------------------|----------------------|---------------|
| Mars, Aldebaran |             4.28 |               168.08 | —             |

Sky map with stellar context

plot_map() draws the conjunction on a Plotly equatorial map with stars from the visible catalogue and constellation names in the field of view (same spirit as Stars.plot_stars). The view is centred on the geometric mean of the body directions and is only produced when in_conjunction is true.

[9]:

conjunction.plot_map(mag_namelimit=5.0)

Loading stellar catalogue montu_stellar_catalogue_v38_visible.csv

Other cases

The examples below reproduce ground-truth conjunctions from the project reference summary. Each case is evaluated explicitly at several observers so you can see how topocentric parallax and local visibility differ from the geocentric geometry.

Mars and Aldebarán

Documented separations for five synodic cycles:

Fecha	Nota
2026-07-13	> 5°
2024-08-04	< 5°
2022-09-07	< 5° (reference)
2021-03-20	> 5°
2019-04-11	> 5°

[10]:

explorer = montu.ConjunctionExplorer(bodies=[mars, aldebaran], maxseparation=8)
conjs = explorer.search(
    start=montu.Time('2019-01-01'),
    end=montu.Time('2027-01-01'),
    observer='geocentric',
)
for conj in conjs:
    print(
        f"Fecha y hora: {conj.mtime.readable.datespice}, "
        f"Separación angular: {conj.separation:.3f}°"
    )

Fecha y hora: 2019-04-15 09:03:00.394556, Separación angular: 6.470°
Fecha y hora: 2021-03-21 06:49:37.896944, Separación angular: 6.944°
Fecha y hora: 2022-09-07 14:28:32.298229, Separación angular: 4.275°
Fecha y hora: 2024-08-04 14:44:14.196480, Separación angular: 4.929°
Fecha y hora: 2026-07-13 01:08:11.100481, Separación angular: 5.313°

Planetary trios

Three planets qualify as a Meeus-style grouping when their maximum pairwise separation is at or below maxseparation.

[11]:

trio_bodies = [
    montu.Planet('Mercury'),
    montu.Planet('Mars'),
    montu.Planet('Saturn'),
]
trio_explorer = montu.ConjunctionExplorer(bodies=trio_bodies, maxseparation=5)
trio_hits = trio_explorer.search(
    start=montu.Time('2026-04-15'),
    end=montu.Time('2026-04-25'),
    observer='geocentric',
)
for hit in trio_hits:
    print(hit.mtime.readable.datespice, f"sep={hit.separation:.2f}°")

2026-04-20 22:41:32.305921 sep=1.65°

[12]:

trio_2026 = montu.Conjunction(
    bodies=trio_bodies,
    maxseparation=5,
    mtime=trio_hits[0].mtime,
    observer='geocentric',
)
trio_2026.show_details()

Conjunction: Mercury–Mars–Saturn
  Epoch (UTC)          : 2026-04-20 22:41:41
  Julian Day (UTC)     : 2461151.445513
  Observer             : geocentric
  Angular separation   : 1.6511° (max allowed 5.0°)
  In conjunction       : yes
  Is visible from site : n/a (geocentric)
  Pair Mercury–Mars
    Separation         : 1.6511°
    Position angle     : 335.74° (N→E)
    Distance           : 1.137052 AU
  Pair Mercury–Saturn
    Separation         : 0.8186°
    Position angle     : 280.09° (N→E)
    Distance           : 9.275015 AU
  Pair Mars–Saturn
    Separation         : 1.3678°
    Position angle     : 185.33° (N→E)
    Distance           : 8.139589 AU
  Mercury
    Phase              : 72.96%
    Angular size       : 0.100 arcmin
    V magnitude        : -0.15
  Mars
    Phase              : 98.08%
    Angular size       : 0.069 arcmin
    V magnitude        : 1.21
  Saturn
    Phase              : 99.96%
    Angular size       : 0.265 arcmin
    V magnitude        : 0.91

[13]:

trio_2026.plot_map()

[14]:

trio_2021_bodies = [
    montu.Planet('Mercury'),
    montu.Planet('Jupiter'),
    montu.Planet('Saturn'),
]
trio_2021_explorer = montu.ConjunctionExplorer(bodies=trio_2021_bodies, maxseparation=5)
trio_2021_hits = trio_2021_explorer.search(
    start=montu.Time('2021-01-05'),
    end=montu.Time('2021-01-15'),
    observer='geocentric',
)
for hit in trio_2021_hits:
    print(hit.mtime.readable.datespice, f"sep={hit.separation:.2f}°")

2021-01-10 12:47:23.104313 sep=2.26°

[15]:

trio_2021 = montu.Conjunction(
    bodies=trio_2021_bodies,
    maxseparation=5,
    mtime=trio_2021_hits[0].mtime,
    observer='geocentric',
)
trio_2021.show_details()
trio_2021.plot_map()

Conjunction: Mercury–Jupiter–Saturn
  Epoch (UTC)          : 2021-01-10 12:47:47
  Julian Day (UTC)     : 2459225.032906
  Observer             : geocentric
  Angular separation   : 2.2590° (max allowed 5.0°)
  In conjunction       : yes
  Is visible from site : n/a (geocentric)
  Pair Mercury–Jupiter
    Separation         : 2.2266°
    Position angle     : 34.51° (N→E)
    Distance           : 4.767688 AU
  Pair Mercury–Saturn
    Separation         : 1.6998°
    Position angle     : 325.75° (N→E)
    Distance           : 9.672860 AU
  Pair Jupiter–Saturn
    Separation         : 2.2590°
    Position angle     : 258.59° (N→E)
    Distance           : 4.916213 AU
  Mercury
    Phase              : 90.88%
    Angular size       : 0.088 arcmin
    V magnitude        : -0.75
  Jupiter
    Phase              : 99.94%
    Angular size       : 0.543 arcmin
    V magnitude        : -1.79
  Saturn
    Phase              : 99.99%
    Angular size       : 0.252 arcmin
    V magnitude        : 0.62

[16]:

trio_2013_bodies = [
    montu.Planet('Venus'),
    montu.Planet('Jupiter'),
    montu.Planet('Mercury'),
]
trio_2013_explorer = montu.ConjunctionExplorer(bodies=trio_2013_bodies, maxseparation=5)
trio_2013_hits = trio_2013_explorer.search(
    start=montu.Time('2013-05-20'),
    end=montu.Time('2013-05-31'),
    observer='geocentric',
)
for hit in trio_2013_hits:
    print(hit.mtime.readable.datespice, f"sep={hit.separation:.2f}°")

2013-05-27 06:42:06.292795 sep=2.36°

[17]:

trio_2013 = montu.Conjunction(
    bodies=trio_2013_bodies,
    maxseparation=5,
    mtime=trio_2013_hits[0].mtime,
    observer='geocentric',
)
trio_2013.show_details()
trio_2013.plot_map()

Conjunction: Venus–Jupiter–Mercury
  Epoch (UTC)          : 2013-05-27 06:42:42
  Julian Day (UTC)     : 2456439.779239
  Observer             : geocentric
  Angular separation   : 2.3634° (max allowed 5.0°)
  In conjunction       : yes
  Is visible from site : n/a (geocentric)
  Pair Venus–Jupiter
    Separation         : 1.7971°
    Position angle     : 118.04° (N→E)
    Distance           : 4.431820 AU
  Pair Venus–Mercury
    Separation         : 2.0285°
    Position angle     : 41.99° (N→E)
    Distance           : 0.507662 AU
  Pair Jupiter–Mercury
    Separation         : 2.3634°
    Position angle     : 355.14° (N→E)
    Distance           : 4.937255 AU
  Venus
    Phase              : 96.30%
    Angular size       : 0.172 arcmin
    V magnitude        : -3.82
  Jupiter
    Phase              : 99.92%
    Angular size       : 0.540 arcmin
    V magnitude        : -1.78
  Mercury
    Phase              : 74.58%
    Angular size       : 0.099 arcmin
    V magnitude        : -0.73

Kepler’s Fire Triangle (autumn 1604)

Mars, Jupiter, and Saturn formed a wide historic grouping. Here we relax the threshold to 10° and search geocentrically through autumn 1604.

[18]:

fire_triangle = montu.ConjunctionExplorer(
    bodies=[montu.Planet('Mars'), montu.Planet('Jupiter'), montu.Planet('Saturn')],
    maxseparation=10,
)
fire_hits = fire_triangle.search(
    start=montu.Time('1604-08-01', calendar='mixed'),
    end=montu.Time('1604-12-31', calendar='mixed'),
    observer='geocentric',
)
for hit in fire_hits:
    print(hit.mtime.readable.datespice, f"sep={hit.separation:.2f}°")

fire_hits[0].plot_map()

1604-09-26 13:59:59.997117 sep=7.55°

[19]:

# Mars, Jupiter, Saturn — February 6 BCE (post triple conjunction)
trio_6bce = montu.ConjunctionExplorer(
    bodies=[montu.Planet('Mars'), montu.Planet('Jupiter'), montu.Planet('Saturn')],
    maxseparation=10,
)
bce_hits = trio_6bce.search(
    start=montu.Time('-0006-02-01', calendar='mixed'),
    end=montu.Time('-0005-03-31', calendar='mixed'),
    observer='geocentric',
)
for hit in bce_hits:
    print(hit.mtime.readable.datespice, f"sep={hit.separation:.2f}°")

bce_hits[0].plot_map()

0006 B.C. 02-18 07:00:00.2880 sep=6.44°

Let’s make a search in 1700 years:

[20]:

# Mars, Jupiter, Saturn — February 6 BCE (post triple conjunction)
explorer = montu.ConjunctionExplorer(
    bodies=[montu.Planet('Mars'), montu.Planet('Jupiter'), montu.Planet('Saturn')],
    maxseparation=5,
)
trio_hits = explorer.search(
    start=montu.Time('-0006-02-01', calendar='mixed'),
    end=montu.Time('1700-03-31', calendar='mixed'),
    observer='geocentric',
    verbose=True,
)
for hit in trio_hits:
    print(hit.mtime.readable.datespice, f"sep={hit.separation:.2f}°")

100%|█████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 623165/623165 [00:31<00:00, 19879.77it/s]

0015 A.D. 02-08 08:28:30.302408 sep=4.66°
0035 A.D. 07-05 06:00:00.00 sep=4.70°
0113 A.D. 06-08 16:59:59.997120 sep=3.25°
0153 A.D. 10-17 00:00:00.00 sep=3.92°
0214 A.D. 06-13 01:59:59.997120 sep=3.65°
0292 A.D. 05-19 09:21:29.701432 sep=3.82°
0471 A.D. 05-07 02:02:20.699512 sep=4.44°
0491 A.D. 03-28 14:56:53.203208 sep=0.56°
0650 A.D. 04-22 18:59:59.994232 sep=4.52°
0670 A.D. 02-26 22:59:59.997120 sep=3.28°
0690 A.D. 08-04 03:00:00.00 sep=4.27°
0828 A.D. 10-12 12:00:00.00 sep=2.25°
0869 A.D. 01-21 15:00:00.00 sep=2.66°
0967 A.D. 05-21 03:59:59.994232 sep=3.54°
1007-09-29 15:00:00.000000 sep=3.58°
1047-12-27 07:35:10.599360 sep=3.71°
1146-05-03 13:00:00.002884 sep=4.16°
1246-11-13 18:00:00.000000 sep=4.70°
1345-03-13 16:59:59.997116 sep=1.98°
1425-10-25 09:00:00.000000 sep=4.84°
1503-10-24 09:00:00.000000 sep=3.06°
1524-02-14 23:00:00.005766 sep=0.94°
1544-02-12 15:00:00.000000 sep=2.27°
1682-09-21 13:00:00.002883 sep=2.34°

Let’s plot the closest:

[21]:

# Escoge la conjunción más cercana de trio_hits y haz un plot
if len(trio_hits) > 0:
    closest = min(trio_hits, key=lambda hit: hit.separation)
    closest.plot_map()
else:
    print("No se encontraron conjunciones en el rango especificado.")

2 planetas y 1 estrella

Mixed groupings combine planets and bright stars. Around 21–22 July 2021, Venus and Regulus close to ~1.1° while Mars stays near the group (~5°). Because Mars–Regulus sits right at that limit, we use maxseparation=5.5 for the three-body search (strict 5° finds no simultaneous minimum).

[22]:

vmr_bodies = [
    montu.Planet('Venus'),
    montu.Planet('Mars'),
    montu.Stars(subset='bright', ProperName='Regulus', return_as='Star'),
]
vmr_explorer = montu.ConjunctionExplorer(bodies=vmr_bodies, maxseparation=10)
vmr_hits = vmr_explorer.search(
    start=montu.Time('2021-07-15'),
    end=montu.Time('2021-07-25'),
    observer='geocentric',
)
for hit in vmr_hits:
    print(hit.mtime.readable.datespice, f"sep={hit.separation:.2f}°")

2021-07-22 03:41:18.798705 sep=4.99°

[23]:

vmr = montu.Conjunction(
    bodies=vmr_bodies,
    maxseparation=5.5,
    mtime=vmr_hits[0].mtime,
    observer='geocentric',
)
vmr.show_details()
vmr.plot_map()

Conjunction: Venus–Mars–Regulus
  Epoch (UTC)          : 2021-07-22 03:41:41
  Julian Day (UTC)     : 2459417.653690
  Observer             : geocentric
  Angular separation   : 4.9933° (max allowed 5.5°)
  In conjunction       : yes
  Is visible from site : n/a (geocentric)
  Pair Venus–Mars
    Separation         : 4.9933°
    Position angle     : 286.06° (N→E)
    Distance           : 1.157413 AU
  Pair Venus–Regulus
    Separation         : 1.0880°
    Position angle     : 202.56° (N→E)
  Pair Mars–Regulus
    Separation         : 4.9885°
    Position angle     : 117.43° (N→E)
  Venus
    Phase              : 84.82%
    Angular size       : 0.205 arcmin
    V magnitude        : -3.85
  Mars
    Phase              : 98.23%
    Angular size       : 0.062 arcmin
    V magnitude        : 1.83
  Regulus
    V magnitude        : 1.36

Triple conjunctions (retrograde loops)

When a faster planet laps a slower one near a stationary point, the separation can reach three local minima within a few months. ConjunctionExplorer.search recovers each crossing.

[24]:

jupiter = montu.Planet('Jupiter')
saturn = montu.Planet('Saturn')

js_explorer = montu.ConjunctionExplorer(bodies=[jupiter, saturn], maxseparation=5)
js_crossings = js_explorer.search(
    start=montu.Time('-0006-01-01', calendar='mixed'),
    end=montu.Time('-0006-12-31', calendar='mixed'),
    observer='geocentric',
)
print(f"Jupiter–Saturn, 7 BCE: {len(js_crossings)} crossings")
for crossing in js_crossings:
    print(f"  {crossing.mtime.readable.datespice}  sep={crossing.separation:.2f}°")

Jupiter–Saturn, 7 BCE: 3 crossings
  0007 B.C. 05-27 05:51:35.804160  sep=0.98°
  0007 B.C. 09-28 18:38:28.296952  sep=0.97°
  0007 B.C. 12-03 09:47:06.498248  sep=1.05°

[25]:

regulus = montu.Stars(subset='bright', ProperName='Regulus', return_as='Star')
jr_explorer = montu.ConjunctionExplorer(bodies=[jupiter, regulus], maxseparation=5)
jr_crossings = jr_explorer.search(
    start=montu.Time('-0003-01-01', calendar='mixed'),
    end=montu.Time('-0001-12-31', calendar='mixed'),
    observer='geocentric',
)
print(f"Jupiter–Regulus, 3–1 BCE: {len(jr_crossings)} crossings")
for crossing in jr_crossings:
    print(f"  {crossing.mtime.readable.datespice}  sep={crossing.separation:.2f}°")

Jupiter–Regulus, 3–1 BCE: 3 crossings
  0003 B.C. 09-12 06:04:04.598392  sep=0.33°
  0002 B.C. 02-15 07:15:23.999040  sep=0.86°
  0002 B.C. 05-07 00:03:33.96952  sep=0.72°

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