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Angular Size

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Back to Math And Terraforming

We don't know how a terraformed planet will look like, but can figure out how would the sky look like. In many cases, like on the moons of Saturn, it will be a spectacle to see all the moons, terraformed or not, shining in the sky.

Basic Formula Edit

If you want to see the size in angles of an object, you must know the distance to it (D), object's diameter (d) and you get the angle (A):

A = 2arctan(d/2D).

This formula is very useful, but there is another one, more easy to compare to objects around us:

Derived Formula Edit

I made a second formula, because using degrees might be tricky and I wanted to compare angular sizes with something everybody has in its room. It is also adapted for Microsoft Excel. Here, the target is an object we want to know how large will look like and observer is the object from where you look. The table will give you two results. First is the maximum size (when target and observer are the closest) and second the minimum size (when target and observer are at opposition).

  • Column A: object name
  • Column B: object's diameter (type values in thousand km)
  • Column C: object's distance (semimajor axis) in million km
  • Column D: observer's semimajor axis (in million km)
  • Column E: determine minimum distance: =ABS(C2-D2)
  • Column F: determine maximum distance: =C2+D2
  • Column G: determine maximum size (seen from minimum distance): =B2/E2
  • Column H: determine minimum size (from maximum distance): =B48/F48

Important notes:

  1. To see the apparent size of an object, do as follows: Suppose you got a result of 9.3. Draw a circle of 9.3 mm diameter on a sheet of paper and look at that circle from a distance of one meter. The more details you can see on the circle, the more things you could see on the targeted object.
  2. The smallest size from where someone could see the object as a small disk, is 0.315. If values are smaller, then you won't be able to see the object as a disk, but as a star. Many people can argue that even below 0.7, you can no longer see an object as a disk.
  3. This calculation method is used for objects orbiting the same celestial body (for example, a system of moons). If you want to see how would you see the planet from a moon, type target's distance zero and observer's distance, moon's semimajor axis. Again, if you want to see how would a moon look like from a planet, type at observer's distance zero and at target's distance the moon's semimajor axis. In this case, you will get two identical values.
  4. When trying to work with objects that have an elliptical orbit, try to use two different measurements: one with apastron/aphelion data (smallest distance) and one with periastron/perihelion data (biggest distance).
  5. If you get a value of less then 0.315, you will see a star and not a disk (or moon). In this case, use the formulas on Magnitude for more data.
  6. If the target is very close, like the Earth seen from a low-orbit satellite, results are not accurate. By sitting on a planet's surface, you will get a value of 2000. For values less then 500, this formula still remains close to reality.

Example in the Solar System Edit

Remember, the units shown here are easy to compare with something around. If an object is visible as a disk of 12.5 units, draw a circle with a diameter of 12.5 mm and look at it from 1 m.

The Sun Edit

The Sun will be seen as a disk from majority of the planets, moons and asteroids, as shown below:

From Mercury: 24.03 units
From Venus: 12.87 units
From Earth: 9.32 units
From Mars: 6.11 units
From Ceres: 3.36 units
From Jupiter: 1.79 units
From Saturn: 0.97 units
From Uranus: 0.48 units
From Neptune: 0.31 units

Note: Starting from Neptune, the Sun will no longer be visible as a disk, but as a star. Also, from no planet, you will see another planet as a disk, even if, at conjunction, Earth from Venus, Venus from Earth, Venus from Mercury and Jupiter from Ceres will be close to the point where you would see them as a tiny disk.

Moons of inner planets Edit

From Earth, you will see the Moon as 9.05 units. From the Moon, the Earth will appear of 33.22 units.

From Mars, you will see its moons as follows:

Phobos from Mars: 2.35 units
Deimos from Mars: 0.55 units
Mars from Phobos: 308 units
Deimos from Phobos: 0.55 units (with a small variation)
Mars from Deimos: 289 units
Phobos from Deimos: 0.94 units

Once terraformed, the planets will no longer look as we know them. They will become blue-white. An interesting fact is that the Moon is separated from Earth by a large enough distance so that you will see both the Earth and Moon. At one AU, at highest separation, the Earth and Moon will be at 2.57 units, so it will be possible to see both.

Jupiter system Edit

At Jupiter, things will be far more interesting:

From Jupiter (suppose you float on top of its atmosphere), you will see the moons as follows:

Almathea - 1.10 units
Io - 8.61 units
Europa - 4.65 units
Ganymede - 4.92 units
Calisto - 4.82 units

From Almathea:

Jupiter - 768 units
Io - 6.01 to 15.14 units
Europa - 3.66 to 6.34 units
Ganymede - 4.20 to 5.93 units
Calisto - 2.33 to 2.83 units

From Io:

Jupiter - 332 units
Almathea - 0.33 to 0.83 units
Europa - 2.86 to 12.52 units
Ganymede - 3.26 to 8.12 units
Calisto - 2.09 to 3.30 units

From Europa:

Jupiter - 208 units
Almathea - 0.5 units
Io - 3.32 to 14.56 units
Ganymede - 3.02 to 13.19 units
Calisto - 1.99 to 4.00 units

From Ganymede:

Jupiter - 130.7 units
Io - 2.43 to 5.60 units
Europa - 1.79 to 7.82 units
Calisto - 1.63 to 5.93 units

From Calisto:

Jupiter - 74.25 units
Io - 1.58 to 2.48 units
Europa - 1.22 to 2.58 units
Ganymede - 1.78 to 6.47 units

All outer moons will not be visible or at best will be seen as stars. Almathea is visible as a disk as far as Europa. All the other small inner moons might be visible as disks from Io. From the outer moons, Jupiter and the Galilean moons are visible. From Himalia, the largest outer moon, you will see something like this:

Jupiter - 12.21 units
Io - 0.31 units
Europa - bright star
Ganymede - 0.45 units
Calisto - 0.40 units. 

As we can see, in Jupiter system, even from the most distant moons, the planet is visible a bit larger then the Moon from Earth and all 4 large moons are visible as bright stars or small disks.

If we think that the four large moons will be terraformed, the view will be more interesting. You could see clouds, oceans and continents somehow similar to how we see 'seas' now on the Moon.

Saturn system Edit

At Saturn, things will be even more interesting.

From Saturn, we will see the moons as follows:

Mimas - 2.14 units
Enceladus - 2.12 units
Tethys - 3.6 units
Dione - 2.98 units
Rhea - 2.90 units
Titan - 4.21 units
Iapetus - 0.41 units

From Mimas, we will see:

Saturn - 630 units
Enceladus - 1.19 to 9.51 units
Tethys - 2.21 to 9.65 units
Dione - 2.00 to 5.84 units
Rhea - 2.14 to 4.46 units
Titan - 3.67 to 4.97 units
Iapetus - 0.42 units

From Enceladus, we could see:

Saturn - 489 units
Mimas - 0.93 to 7.48 units
Tethys - 1.99 to 18.63 units
Dione - 1.49 to 8.08 units
Rhea - 1.99 to 5.28 units
Titan - 3.53 to 5.23 units
Iapetus - 0.38 to 0.44 units

From Tethys, the sky will look like this:

Saturn - 395 units
Mimas - 0.82 to 3.63 units
Enceladus - 0.95 to 8.85 units
Dione - 1.67 to 13.70 units
Rhea - 1.86 to 6.58 units
Titan - 3.39 to 5.55 units
Iapetus - 0.38 to 0.45 units

From Dione, this is what you will see:

Saturn - 309 units
Mimas - 0.71 to 2.06 units
Enceladus - 0.82 to 3.63 units
Tethys - 1.58 to 12.95 units
Rhea - 1.69 to 10.18 units
Titan - 3.22 to 6.09 units
Iapetus - 0.37 to 0.46 units

From Rhea:

Saturn - 221 units
Mimas - 0.56 to 1.16 units
Enceladus - 0.66 to 1.74 units
Tethys - 1.29 to 4.58 units
Dione - 1.24 to 7.49 units
Titan - 2.94 to 7.41 units
Iapetus - 0.36 to 0.48 units

From Titan (the best candidate for terraforming):

Saturn - 95.3 units
Mimas - star to 0.38 units
Enceladus - 0.34 to 0.51 units
Tethys - 0.70 to 1.15 units
Dione - 0.70 to 1.33 units
Rhea - 0.87 to 2.20 units
Hyperion - star to 1.04 units
Iapetus - 0.31 to 0.36 units

From Iapetus:

Saturn - 32.71 units
Tethys - star to 0.33 units
Dione - star to 0.35 units
Rhea - 0.37 to 0.53 units
Titan - 1.07 to 2.20 units

From Phoebe:

Saturn - 9.05 units
Titan - 0.37 to 0.44 units

What is to be noticed is that in Saturnian System, moons are seen much smaller then in the Jovian System. Not all of them are suitable for terraforming, so their appearance will remain almost the same. Phoebe is visible from the entire system as a star, while all moons except Titan are visible as stars from Phoebe. The rings rotate in the same plane with the inner moons, so they will not be clearly visible from nearby.

Uranus System Edit

The simulation is based on Uranus and its 5 large moons:

From Uranus, you would see:

Miranda - 3.66 units
Ariel - 7.07 units
Umbriel - 5.17 units
Titania - 3.61 units
Oberon - 2.61 units

From Miranda, we could see:

Uranus - 393 units
Ariel - 4.23 to 21.82 units
Umbriel - 3.29 to 12.05 units
Titania - 2.79 to 5.13 units
Oberon - 2.14 to 3.35 units

From Ariel, we would see:

Uranus - 266 units
Miranda - 1.48 to 7.21 units
Umbriel - 2.80 to 33.40 units
Titania - 2.15 to 6.43 units
Oberon - 1.97 to 3.88 units

From Umbriel, the sky would look like this:

Uranus - 225 units
Miranda - 1.34 to 4.36 units
Ariel - 3.24 to 38.66 units
Titania - 2.38 to 7.50 units
Oberon - 1.88 to 4.26 units

From Titania, it will be like this:

Uranus - 116 units
Miranda - 0.84 to 1.54 units
Ariel - 2.16 to 5.52 units
Umbriel - 1.77 to 5.57 units
Oberon - 1.49 to 10.35 units

From Oberon, these are the estimates:

Uranus - 87 units
Miranda - 0.66 to 1.04 units
Ariel - 1.75 to 3.45 units
Umbriel - 1.44 to 3.27 units
Titania - 1.55 to 10.72 units. 

What is interesting in the Uranian System is that the moons sometimes come very close one to each other. So, at opposition, sometimes you can barely see them as disks, while at conjunction you can see them sometimes larger then Earth's Moon.

Neptune System Edit

I only included the most known moons of Neptune: Proteus, Triton and Nereid.

From Neptune:

Proteus - 3.56 units
Triton - 7.62 units
Nereid - pale star to bright star

From Proteus:

Neptune - 417 units
Triton - 5.72 to 11.41 units
Nereid - pale star to bright star

From Triton:

Neptune - 139 units
Proteus - 1.77 units
Nereid - pale star to 0.33 units

From Nereid:

Neptune - 5.10 to 35.89 units
Proteus - pale star to 0.33 units
Triton - star to 2.66 units. 

The Neptunian System is interesting because we have retrograde and highly elliptic bodies. If one day Triton will be terraformed, on its surface, you will see a sunrise in West and a sunset in East, while on Nereid, Neptune will be sometimes larger and sometimes smaller.

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