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Back to Math And Terraforming

B - type star.jpeg

For celestial bodies large enough to be seen as crescents or disks (like moons in a planetary system), please go to Angular Size.

The magnitude of a celestial body is the luminosity (in all wavelengths, on a specified wavelength or in visible light) of a celestial body[1].

The following formulas are made for Microsoft Excel and are a good tool to see how would you see a planet or moon from a certain point. Please note that the formulas shown below are not token from a science magazine, they are derived by myself from existing formulas on Wikipedia. So, if you know better formulas, please improve.

The first thing you need to know is the star's Solar Constant(DKs). From this, you can calculate the solar constant for a specified planet (Ks). Then, if you know the diameter and albedo, you can determine the amount of radiation coming from the target planet. Finally, knowing the distance from you to the planet, you can find out the magnitude.

I calibrated the magnitude scale on Neptune, because it has the smallest magnitude variation from all planets. This is because of two things. First, Neptune doesn't change distance to Earth too much. And second, it doesn't change its albedo too much.

Formula

The following example is for Microsoft Excel. It shows how Neptune would be seen from Earth:

  • Column A - type target's name (planet you want to see) - Neptune
  • Column B - type target's distance to the Sun in million km - 4501.445
  • Column C - type distance to the observer (in this case, Neptune to Earth, is the same as Neptune to the Sun) - 4501.445
  • Column D - type target's diameter in thousands km (for Neptune is 49.324)
  • Column E - type Solar Constant (for this simulation it is 44235)
  • Column F - type target's albedo (for Neptune, it is 0.367)
  • Column G - type magnitude formula: =-2.5*(LOG10(((((E2/B2/B2)/F2)*D2*D2)/C2/C2)/(0.000000714182)))+7.9

Please note that the formula is for row 2, suppose all data is inserted on row 2. For next rows, just copy it from row 2.

If you apply these formulas, you will get for Neptune the magnitude of 7.9. Since the average human eye cannot see objects with a higher magnitude of 6, you can see why Neptune is invisible from Earth. However, if you want to have fun, place Neptune closer to the Sun, increase the solar constant or place the observer closer to Neptune and there you go, it becomes visible.

In a similar way, if you want to see the opposite, Earth from Neptune, you can replace the values (keeping distance to the observer the same) and you will get a magnitude of 3.44. So, you can see the Earth from Neptune. This is true, because the Earth receives far more light from the Sun (and radiates more light), despite its much smaller size.

On the other hand, if you try to see the data for the hypothetical Planet Nine, suppose it has the same diameter as Saturn (116000 km) and the same albedo (0.47), you will get for a distance of 700 AU, a magnitude of 20. Only powerful telescopes can spot something like that. But, if you try to see Saturn from Planet Nine, you get a magnitude of 10.7.

Stellar Distances

There are 3 parameters strongly correlated one to each other. The first is absolute magnitude (the luminosity of a star as viewed from 10 parsecs or 32.59 light years), the second is apparent magnitude (how bright is the star as seen from the observer) and the third is distance (measured in parsecs). We will note these as follows:

  • M is absolute magnitude
  • m is apparent magnitude
  • D is distance in parsecs, while d is distance in light years;

The basic formula is: m-M = 5(lgD-1).

Simulation For Solar System

For celestial bodies that are visible as disks and moons, please see Angular Size.

Basic data: The following scheme contains distance to parent body (millions km), diameter (thousands km) and albedo for major celestial bodies in the Solar System.

Planets

From each planet or moon, other celestial bodies will appear having the following magnitude:

Mercury

  • Venus: -5.3 to -2.7
  • Earth: -2.8 to -1.1
    • Moon: 1.1 to 2.8
  • Mars: 1.6 to 2.8
  • Jupiter: 0 to 0.3
  • Saturn: 2.8 to 3.0

Venus

  • Mercury: -2.5 to 0.2
  • Earth: -4.6 to -0.6
    • Moon: -0.7 to 3.3
  • Mars: 0.9 to 3.1
  • Jupiter: -0.2 to 0.4
  • Saturn: 2.7 to 3.1

Earth

  • Mercury: -1.2 to 0.5
  • Venus: -5.8 to -1.8
  • Mars: -0.1 to 3.4
  • Jupiter: -0.3 to 0.5
  • Saturn: 2.7 to 3.1

Mars

  • Mercury: 0.1 to 1.2
  • Venus: -3.5 to -1.3
  • Earth: -3.2 to 0.3
    • Moon: 0.7 to 4.1
  • Jupiter: -0.6 to 0.7
  • Saturn: 2.5 to 3.2

Ceres

  • Mercury: 1.7 to 2.3
  • Venus: -1.4 to -0.3
  • Earth: -0.5 to 1.1
    • Moon: 3.4 to 5.0
  • Mars: 1.8 to 4.5
  • Jupiter: -1.5 to 1.1
  • Saturn: 2.1 to 3.5

Jupiter

  • Mercury: 3.2 to 3.6
  • Venus: 0.2 to 0.8
  • Earth: 1.3 to 2.2
    • Moon: 5.2 to >6
  • Mars: 4.2 to 5.5
  • Saturn: 1.2 to 3.8

Saturn

  • Mercury: 4.6 to 4.8
  • Venus: 1.7 to 2.0
  • Earth: 2.9 to 3.4
  • Mars: 5.9 to >6
  • Jupiter: -0.2 to 2.4

Uranus

  • Mercury: 6.2 to 6.3
  • Venus: 3.3 to 3.5
  • Earth: 4.5 to 4.8
  • Mars: 5.9 to >6
  • Jupiter: 2.3 to 3.5
  • Saturn: 2.9 to 5.3

Neptune

  • Venus: 4.3 to 4.4
  • Earth: 5.5 to 5.7
  • Jupiter: 3.5 to 4.3
  • Saturn: 4.6 to 6.0

Pluto

  • Venus: 4.9 to 5.0
  • Jupiter: 4.2 to 4.8
  • Saturn: 5.4 to >6

Important notes:

  1. These numbers show the visual magnitude as seen at complete illumination (phase angle = 0). In practice, values are smaller, since a celestial body cannot be 100% illuminated.
  2. In case of an inner planet, as seen from an outer planet, it can never be fully illuminated (phase angle zero). In that case, it should actually be at opposition with the Sun, so it will not be visible. Because of this, when you look at an inner planet, the real values will be by one order of magnitude smaller.
  3. If a planet is too close to the Sun, Sun's luminosity will render that planet invisible. For example, in theory, Jupiter should be visible for Pluto, but in practice, that will never happen.
  4. When looking for a satellite, it is visible only if it is far enough from its parent body. For example, the Moon will be visible from Mercury, Venus or Mars, only when it is far enough from Earth.
  5. In this calculation, extra brightness from rings is not included.
  6. Once terraforming will occur, many celestial bodies will change their albedo, modifying their brightness. So will happen with their color. For example, Venus and Mars will become blue, like the Earth. Venus will be dimmer and Mars brighter then they are today.

Moons

In most cases, from one moon, the other moons will be visible as small disks. Please go to Angular Size for details.

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