## The Small-Angle Approximation

Relatively complex geometry is necessary to determine the true size of distant objects based on their apparent size as seen by an observer.

In this case, we are determining the opposite; we know the true diameter of the object in the sky and we know its distance from the observer, so we can use a shortcut to determine how large the object appears to be in the sky.

The following equation uses absolute values to return the apparent diameter in the sky in radians:

In this case, we are determining the opposite; we know the true diameter of the object in the sky and we know its distance from the observer, so we can use a shortcut to determine how large the object appears to be in the sky.

The following equation uses absolute values to return the apparent diameter in the sky in radians:

Another version of this equation returns the diameter in the sky in arcseconds (1/3600˚):

## Example: The Angular Diameter of The Moon

Actual Diameter Of The Moon: 3474.2 km

Actual (Average) Distance To The Moon: 384399 km

Actual (Average) Distance To The Moon: 384399 km

… multiplied by 57.2958 gives us the answer in degrees:

## Example: The Angular Diameter of the Sun

Actual Diameter Of The Sun: 1391400 km

Actual (Average) Distance To The Sun: 1.496E+8 km

Actual (Average) Distance To The Sun: 1.496E+8 km

… multiplied by 57.2958 gives us the answer in degrees:

## Apparent Diameters and Eclipses

Note that our answer for the Moon and for the Sun are very close to one another. This explains why total eclipses of the Sun by the Moon are possible; from the vantage point of the Earth, both look more-or-less the same size in the sky, so the Moon can completely cover the Sun when it passes between it and the Earth.

Note that the Earth's orbit around the Sun is an ellipse, and as a result, the distance to the Sun changes slightly over the period of a year, so the angular diameter of the Sun varies from about 0.527˚ to about 0.545˚. Similarly, because the Moon's orbit around the Earth is also an ellipse, the angular diameter of the Moon also varies, from about 0.488˚ about 0.568˚.

Thus, there are times when the Moon does not completely cover the Sun, leading not to a total eclipse, but to an annular eclipse (from the Latin "annulus", meaning a ring, because a ring of Sun remains visible around the outside of the dark disk of the Moon).

Note that the Earth's orbit around the Sun is an ellipse, and as a result, the distance to the Sun changes slightly over the period of a year, so the angular diameter of the Sun varies from about 0.527˚ to about 0.545˚. Similarly, because the Moon's orbit around the Earth is also an ellipse, the angular diameter of the Moon also varies, from about 0.488˚ about 0.568˚.

Thus, there are times when the Moon does not completely cover the Sun, leading not to a total eclipse, but to an annular eclipse (from the Latin "annulus", meaning a ring, because a ring of Sun remains visible around the outside of the dark disk of the Moon).

## Apparent Brightness (of Non-Luminant Objects)

The means for calculating the apparent brightness of

*non-luminant objects*[1] is rather more complicated than calculating the apparent brightness of stars (discussed elsewhere). We must take into account the reflectivity (*albedo*) of the reflecting object, and the relative distances between the object, the source illuminating it, and its distance to the observer.## Example: The Apparent Brightness Of The Moon

To do this calculation, we need to know a few vital facts about the Moon:

which is about 0.08% as bright as a 60 watt light bulb.

## Fictional Albedos

"How do I calculate the albedo of my fictional planets?" one may ask.

Well....

It's not straightforward, unfortunately. Albedo is the ratio of the amount light reflecting from an object divided by the amount of light falling on it (the

The radius of the body is a factor in the equation above; such that, if Planet A is twice the diameter of Planet B, but they both have the same albedo, Planet A will appear brighter than Planet B when observed from the same distance. Conversely, if Planet A and Planet B are the same size, but Planet A has twice the albedo of Planet B, Planet A will again appear brighter than Planet B when viewed from the same distance. A real-world example is that of Earth and Venus: Venus is 94.99% the radius of the Earth, but its albedo is 0.76, compared to Earth's albedo of 0.40. Earth is larger, but viewed from the same distance, it would be dimmer than Venus.

Albedo is a function of the material the body is made of; for planets, prime considerations will be atmospheres. This helps explain the Earth-Venus example above--

When it comes to fictional planets, my best advice is to

For instance, let's say we have a planet—Taiar—which is a Midgean, with a surface similar to Mercury. Mercury's albedo is 0.06. Let's say Taiar is 0.6213 times Mercury's radius. Thus:

Well....

It's not straightforward, unfortunately. Albedo is the ratio of the amount light reflecting from an object divided by the amount of light falling on it (the

*incident*light). If it reflects all of the incident light, it has an albedo of 1.0; if it reflects none of the incident light, it has an albedo of 0.0; and so on.The radius of the body is a factor in the equation above; such that, if Planet A is twice the diameter of Planet B, but they both have the same albedo, Planet A will appear brighter than Planet B when observed from the same distance. Conversely, if Planet A and Planet B are the same size, but Planet A has twice the albedo of Planet B, Planet A will again appear brighter than Planet B when viewed from the same distance. A real-world example is that of Earth and Venus: Venus is 94.99% the radius of the Earth, but its albedo is 0.76, compared to Earth's albedo of 0.40. Earth is larger, but viewed from the same distance, it would be dimmer than Venus.

Albedo is a function of the material the body is made of; for planets, prime considerations will be atmospheres. This helps explain the Earth-Venus example above--

*all*of the light reflecting from Venus is reflected off its atmosphere, but some of the light reflecting from Earth is reflecting from clouds, ocean, land surface, ice, etc.When it comes to fictional planets, my best advice is to

- Find the known albedo of a Solar System object which is most similar to your fictional planet;
- Take the ratio of your planet's radius compared to that of the example planet;
- Multiply the albedo of the known planet by the ratio calculated in Step 2.

For instance, let's say we have a planet—Taiar—which is a Midgean, with a surface similar to Mercury. Mercury's albedo is 0.06. Let's say Taiar is 0.6213 times Mercury's radius. Thus:

... so, we'd say that Taiar's albedo is 0.037278.

This, of course, assumes that Taiar, as Mercury, has no atmosphere, and that Taiar's surface is composed of more-or-less the same materials as the surface of Mercury.

If, on the other hand, Taiar is 62.13% of the

This, of course, assumes that Taiar, as Mercury, has no atmosphere, and that Taiar's surface is composed of more-or-less the same materials as the surface of Mercury.

If, on the other hand, Taiar is 62.13% of the

*size*of Mercury, but has an*atmosphere*like Titan's, it would be better to use Titan's albedo as the standard. Since Mercury is 0.9262 times the size of Titan (yep—Titan is larger than Mercury), and Taiar is 62.13% the size of Mercury, then Taiar is 0.57556 times Titan's diameter. Titan's albedo is 0.22, so the equation becomes:... and this is the value we'd use in the Apparent Brightness equation above.

Is this a rigorously scientific calculation? No, and so I hereby invoke The Omega Argument.

Is this a rigorously scientific calculation? No, and so I hereby invoke The Omega Argument.

## The naked-Eye Limit

From Earth, with minimal light pollution in the atmosphere, Saturn is the last of the naked-eye planets (assuming we mean a

Saturn's apparent brightness (on average) is 1.338E-7 w/cm³, so it makes sense to say that no object with a calculated apparent brightness less than this value is a naked-eye object.

Note that this is NOT saying that no object orbiting farther from its star than does Saturn from the Sun is visible. Apparent brightness is not just a function of distance from the illuminating body, but also of the size and reflectivity of the object, and its distance from the observer.

For instance, if Jupiter were moved to Saturn's orbit, it would be almost 1.5 times brighter than Saturn is as the same distance, and certainly remain a naked-eye object. Uranus, however, moved inward to Saturn's orbit would still have only 17% of Saturn's brightness and remain invisible to the naked eye.

*human*eye), meaning that it can be seen in the night sky with the eye alone. Uranus and Neptune—as well as the dwarf planets and other TNO/Kuiper Belt objects—do populate the Solar System beyond the orbit of Saturn, but they are not visible even to the most eagle-eyed of humans without the aid of binoculars and/or telescopes.Saturn's apparent brightness (on average) is 1.338E-7 w/cm³, so it makes sense to say that no object with a calculated apparent brightness less than this value is a naked-eye object.

Note that this is NOT saying that no object orbiting farther from its star than does Saturn from the Sun is visible. Apparent brightness is not just a function of distance from the illuminating body, but also of the size and reflectivity of the object, and its distance from the observer.

For instance, if Jupiter were moved to Saturn's orbit, it would be almost 1.5 times brighter than Saturn is as the same distance, and certainly remain a naked-eye object. Uranus, however, moved inward to Saturn's orbit would still have only 17% of Saturn's brightness and remain invisible to the naked eye.

1. The apparent brightness of luminant objects (stars) will be covered in a later blog.