A&B's Astronomy Lab, Fall 2001, No. 3

Measuring Angular and Physical Sizes

The diameter of the Sun is over 400 times larger than the diameter of the Moon.
Then how can they appear to be the same size? The sky does some magic?
By some measurement, you will figure it out why!

Understanding Angular Size

The angluar size is an angle subtended by an object on the sky. The angle may be measured in units of radians, or in units of degrees, arc minutes (1 degree = 60 arcminutes), and arc seconds (one arc minute = 60 arc seconds). For example, the angular size of the moon in the sky is 30 arc minutes. The human eye cannot resolve details smaller than a few arc minutes. The best telescopes in space (e.g. Hubble Space Telescope) can resolve details as small as a hundreth of an arcsecond. You can use a protractor, but it's useless under the real sky. But as you can see in the figure below, you can use your body to measure the angular size of an object on the sky.

Now, can you measure the angular sizes of the George Washington Bridge which you can see on the roof of Pupin or the Moon on the sky? However, since the length and size of your arms and hands might be different from the "standard size", let's measure the exact angular sizes of your thumb, fist, palm, and so on.
Now, use your body protractor, measure the angle substanded by the Moon on the sky.

How do we measure actual sizes from angular size?

Imagine, Pooh is playing with one of his friends, Piglet (Personally, I think Piglet is a very unrealistic character. I didn't even know that this character is a pig before someone let me know his name and its origin...anyway!). If watch them sitting next to each other, i.e. they are at the same distance from you, you know Pooh seems to be larger than Piglet because the actual size of Pooh is larger than Piglet. However, if Pooh left Piglet to bring his honey, Piglet could appear larger than Pooh depending on the distances, that is, someone who's never seen the cartoon might be suprised to see that Pooh is taller than Piglet when Pooh comes back with honey. Like this, the apparent size changes with the distance between the object itself and the observer.

Think about three different cases,
[Case 1] For a given distance:
Larger actual size gives you larger angular size
Smaller actual size gives you smaller angular size
(you know that bigger things look bigger!)
[Case 2] For a given actual size:
Greater distance gives you smaller angular size
Shorter distance gives you larger angular size
(you know things look smaller when they are farther away!)
[Case 3] For a given angular size
Greater distance implies larger actual size
Shorter distance implies Smaller actual size
(think about this one for a minute. hold a toy close enough to your eye, and it fills as much of your vision as the real thing)

Appendix - Convenient Distance Units in Astronomy
AU (Astronomical Unit), Light Year, and pc (parsec).

AU (Astronomical Unit) is the average distance of the Earth from the Sun (1 AU ∼ 150 million km). Light Year is the distance that light can travel in year, which is 9.46 trillion km. The most direct way to measure the distances to stars is with stellar parallax, the small annual shift in a star's apparent position caused by the Earth's motion around the Sun. Astronomers measure stellar parallax by comparing observations of a nearby star made 6 months apart (see the figure). The nearby star apprears to shift against the background of more distant stars because we are observing it from two opposite points of the Earth's orbit. The star's parallax angle is defined as half the star's annual back-and-forth shift. By definition, the distance to an object with a parallax angle of 1 arcsecond is one parsec (pc). Since even nearby stars are still located at far distances compared to the baseline (the distance between the Sun and the Earth), the parallax angle is so small (e.g. for the nearest star, Proxima Centauri, the parallax angle is only 0.77 arcsecond) and it's hard to measure. Therefore, measuring parallax is difficult (because the angle is extremely small!) and this method to measure the distance to the star is limited to the stars in solar neighborhood.
d (in parsec) = 1/[p (arcseconds)]

Understanding Angular Size
The angluar size is an angle subtended by an object on the sky. The angle may be measured in units of radians, or in units of degrees, arc minutes (1 degree = 60 arcminutes), and arc seconds (one arc minute = 60 arc seconds). For example, the angular size of the moon in the sky is 30 arc minutes. The human eye cannot resolve details smaller than a few arc minutes. The best telescopes in space (e.g. Hubble Space Telescope) can resolve details as small as a hundreth of an arcsecond. You can use a protractor, but it's useless under the real sky. But as you can see in the figure below, you can use your body to measure the angular size of an object on the sky.

Now, can you measure the angular sizes of the George Washington Bridge which you can see on the roof of Pupin or the Moon on the sky? However, since the length and size of your arms and hands might be different from the "standard size", let's measure the exact angular sizes of your thumb, fist, palm, and so on. Now, use your body protractor, measure the angle substanded by the Moon on the sky.

How do we measure actual sizes from angular size?
Imagine, Pooh is playing with one of his friends, Piglet (Personally, I think Piglet is a very unrealistic character. I didn't even know that this character is a pig before someone let me know his name and its origin...anyway!). If watch them sitting next to each other, i.e. they are at the same distance from you, you know Pooh seems to be larger than Piglet because the actual size of Pooh is larger than Piglet. However, if Pooh left Piglet to bring his honey, Piglet could appear larger than Pooh depending on the distances, that is, someone who's never seen the cartoon might be suprised to see that Pooh is taller than Piglet when Pooh comes back with honey. Like this, the apparent size changes with the distance between the object itself and the observer.
	Think about three different cases, [Case 1] For a given distance: Larger actual size gives you larger angular size Smaller actual size gives you smaller angular size (you know that bigger things look bigger!) [Case 2] For a given actual size: Greater distance gives you smaller angular size Shorter distance gives you larger angular size (you know things look smaller when they are farther away!) [Case 3] For a given angular size Greater distance implies larger actual size Shorter distance implies Smaller actual size (think about this one for a minute. hold a toy close enough to your eye, and it fills as much of your vision as the real thing)