Parallax, angular size, and angular resolution
Exercise 4.3. Repeat the two previous exercises using the diameter of the Earth’s orbit as your baseline
4.2 Angular size
Every child knows that an object “gets smaller” as it gets farther away. Yet the physical size of the object is not changing, so what makes it appear smaller?
The answer is that the “angular size” of the object decreases with distance, and angular size is the subject of this section.
4.2.1 Angular-size concept
To understand why angular size decreases with distance you have to realize that the angular size (also called the “angular diameter”) of an object for a given observer is the angle between the line of sight from the observer to one edge of the object and the line of sight to the opposite edge of the object. This is shown in Figure 4.4.
4.2 Angular size 107
Observer Angular
size Celestial
object
Distance Physical
size of object
Figure 4.4 Angular size of a celestial object.
Observer Larger
angular size Same-sized
object
Smaller distance Physical
size of object
Figure 4.5 Angular-size dependence on distance.
Some astronomy students have trouble differentiating angular size from par-allax angle, but you can understand the difference by comparing Figure 4.4 to Figure 4.1. Both angular size and parallax involve the difference between two lines of sight, but the way those lines of sight are formed is completely differ-ent. The key difference is that parallax occurs when the same object is observed from two different locations (the ends of the baseline, as shown in Figure 4.1), while angular-size measurement is made from the same location, but two dif-ferent points on the object are observed (as shown in Figure 4.4). Put simply, the observation point changes for parallax measurements, but the observation point remains the same for angular-size measurements.
A tip that some students find helpful for differentiating angular size from parallax is to remember which way the triangle points. In parallax measure-ments, the baseline (the base of the triangle) is between the two observing locations, and the triangle points away from the observer. In angular-size mea-surements, the base of the triangle is the physical size of the object, and the triangle points toward the observer.
Unlike physical size, which is completely determined by the dimensions of the object, angular size depends on both the physical size of the object and the distance to the observer. To see that, consider what would happen if the object in Figure 4.4 were closer to the observer, as shown in Figure 4.5.
In this case, the smaller distance to the object would increase the angle formed at the observation point by the two lines of sight to opposite edges of
108 Parallax, angular size, and angular resolution
Observer Smaller
angular size Smaller
object
Same distance Physical
size of object
Figure 4.6 Angular-size dependence on physical size.
the object. Thus angular size is inversely related to distance; for small angles, cutting the distance in half doubles the angular size of the object (so angular size is inversely proportional to distance for small angles).
As you can see in Figure 4.6, angular size also depends on the physical size of the object – larger objects subtend a greater angle than smaller objects at the same distance. And as you may have guessed, for small angles, angular size is directly proportional to physical size. Combining the effects of distance and physical size, the angular size of an object may be written as
angular size∝
physical size distance
(4.4) or
angular size= (const) ×
physical size distance
. (4.5)
4.2.2 Calculating angular size
The constant of proportionality in Eq. 4.5 is 1.0 as long as the units of the angu-lar size are radians and the units of the physical size are the same as the units of the distance. But even without knowing the units, you can use the propor-tional relationship of Eq. 4.4 or 4.5 to solve angular-size problems using the ratio method. Here’s an example.
Example: The Sun’s diameter is approximately 1,390,000 km, which is about 400 times larger than the Moon’s diameter. But the Sun’s distance from Earth is also about 400 times larger than the Moon’s distance, so how does the angular size of the Sun compare to the angular size of the Moon, as seen from Earth?
As in the shark–parallax example, you can use the ratio method to solve this problem in your head. Since physical size is in the numerator of Eq. 4.4 and distance is in the denominator, the Sun’s greater size will be offset by its greater
4.2 Angular size 109
distance – if both numerator and denominator are 400 times larger for the Sun compared to the Moon, then the angular size of the Sun and the Moon must be approximately equal. That’s why the Moon can just barely cover the Sun during a total solar eclipse.2
Here’s a version of the angular-size equation that reminds you of the units:
angular size (rad)=physical size (same units as distance)
distance (same units as physical size). (4.6) Example: What is the diameter of a planet whose angular size is 47as seen from the Earth when the distance to the planet is 4.2 AU?
In this problem, you’re given the angular size of an object and the distance from the observer to the object, and you’re asked to find the physical size of the object. These are the variables in Eq. 4.6, although you’ll have to do some unit conversion before you can use that equation. And, as always, it’s a good idea to first rearrange the equation to move the quantity you’re after onto the left side:
angular size= physical size distance
physical size= (angular size)(distance). (4.7) Before plugging in the values for angular size and distance, you’ll have to convert the units of angular size from arcseconds to radians, using the fact that 1◦↔ 3,600:
angular size= 47×
1 degree 3,600
×
π radians 180 degrees
= 2.28 × 10−4rad.
And since the units of AU work well for interplanetary distances but are gen-erally not convenient for expressing the diameter of a planet, converting the distance of 4.2 AU to kilometers is also a good idea:
distance= 4.2 AU ×
1.5 × 108km 1 AU
= 6.3 × 108km.
2 Since the Moon’s orbit around the Earth and the Earth’s orbit around the Sun are elliptical rather than circular, the Earth–Moon and Earth–Sun distances change slightly over time. This gives rise to “annular” solar eclipses in which the Moon’s angular size is smaller than the Sun’s so the Sun is not entirely covered.
110 Parallax, angular size, and angular resolution
With the parameters in the correct units, you can now plug the values into Eq. 4.7:
physical size= (angular size)(distance) = (2.28 × 10−4rad)(6.3 × 108km)
= 143,640 km,
which is the diameter of the planet Jupiter.
If you want to find the angular size of an object in degrees, you can use Eq. 4.6 to find the angular size in radians and then convert your answer to degrees, or you can include the conversion factor in the angular-size equation like this:
angular size (deg)=
180 deg π
× angular size (rad)
=
180 deg π
×physical size distance or
angular size (deg)= 57.3◦
physical size distance
, (4.8)
in which the units of the object’s physical size must be the same as the units of the distance.
The next section of this chapter will help you understand why the very small angular size of the stars makes it virtually impossible to resolve their surfaces even with the largest telescopes currently available. But first, here’s a chance to exercise your understanding of angular size.
Exercise 4.4. What is the angular size of the star Betelgeuse as seen from