Parallax, angular size, and angular resolution
Exercise 4.6. Calculate the angular resolution of the human eye for visible light for a dilated pupil with a diameter of 9 mm
(4.14) Here are two exercises that will help you check your understanding of the equation for angular resolution.
Exercise 4.5. The undilated pupil of a typical human eye has a diameter of about 4 mm. Calculate the angular resolution of the human eye for visible light.
Exercise 4.6. Calculate the angular resolution of the human eye for visible light for a dilated pupil with a diameter of 9 mm.
4.4 Chapter problems
4.1 What is the parallax angle of the planet Venus using Earth’s diameter as baseline when Venus is 0.5 AU from Earth?
4.2 How far away is a planet with a parallax angle of 4.6when a baseline of 1,000 km is used?
4.3 What length of baseline would produce a parallax angle of one arcsecond for the Andromeda Galaxy at a distance of 2.5 million light years from Earth?
4.4 Jupiter’s moon Io orbits at a distance of about 420,000 km from Jupiter.
What is the angular size of Io’s orbit when Jupiter (and Io) are 4.2 AU from Earth?
4.5 What is the angular size of the moon Triton’s orbit around Neptune as seen from Earth when Neptune is at closest approach to Earth? Triton’s
4.4 Chapter problems 121
orbital period around Neptune is 5.9 Earth days, and Neptune’s orbital period around the Sun is about 164 Earth years.
4.6 How far away is a galaxy with physical size (diameter) of 100,000 light years and angular size of 12 arcminutes?
4.7 What is the angular resolution in visible light of the Keck telescope, which has a mirror with diameter of 33 feet?
4.8 What is the angular resolution of the 100-meter Greenbank radio tele-scope when operated at a frequency of 400 MHz?
4.9 How large would a radio telescope operating at 1 GHz have to be to achieve the same resolution as the 200-inch Palomar telescope at visible light?
4.10 Imagine an extrasolar planet orbiting a 3-solar-mass star once every 75 Earth days. If the star is 10 light years from Earth, how large would a visible-light telescope have to be to resolve the planet from its parent star?
5
Stars
The sight of the starry night sky has been inspiring humans for thousands of years. For most of our history, the nature of those shimmering jewels has been a mystery, thought by many to be forever beyond our understanding. But in the last two hundred years, we’ve learned how to extract information as well as inspiration from starlight. By taking that light apart using spectrographs and making precise measurements of its brightness, we’ve come to understand a great deal not only about the nature of stars, but also about the structure and workings of the Universe at large. The mathematics behind that understanding is the subject of this chapter.
5.1 Stellar parallax
As described in Section 4.1, parallax is an apparent shift of an object’s posi-tion due to the changing line of sight between the observer and the object. The amount of that shift depends on the distance to the object and on how far the observer moves (called the baseline of the measurement). Astronomers take advantage of this effect as Earth moves around in its orbit to measure distances to nearby stars, which appear to shift position against distant background stars.
5.1.1 Stellar parallax equation
As the Earth moves from one side of its orbit to the other over the course of half a year, a star’s resulting parallax angular shift and its distance from Earth are related by this equation:
d = 1
p, (5.1)
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5.1 Stellar parallax 123
where d is the distance to the star in units of parsecs (pc), and p is the parallax angular shift in units of arcseconds (). One parsec is a huge distance compared to human experience (1 pc= 3.09 × 1013 km= 3.26 light years), but it is relatively small in astronomy, which frequently deals with immense distances.
Likewise, 1 arcsecond is a tiny angle compared to human experience (1 = 1/3,600th of 1 degree), but it is useful in astronomy, which frequently deals with very tiny angles.
If you want to see how this equation comes about from the general parallax Eq. 4.1, take a look at the problems at the end of this chapter and the on-line solutions. If you do that, you’ll see that the angle used in Eq. 5.1 is not the full parallax angle shown in Figure 4.1, but rather half of that angle. Most astronomy texts use the term “parallax angle” or “parallax angular shift” to refer to the angle that is half of the full parallax angle shown in Figure 4.1, so we’ll do the same.
In order for Eq. 5.1 to work, the quantities must always be in the specified units. In fact, the distance unit of 1 parsec is defined as the distance from Earth to a star which shows a parallax angular shift of 1 arcsecond as viewed from opposite sides of Earth’s orbit. If you take care to ensure that your quantities are always in these prescribed units, then this equation is just an inverse pro-portionality with no constants, making it one of the simplest you will encounter in astronomy.
Example: Below are the parallax angles for four stars. Which of these stars is farthest from Earth, and which is closest?
Alcor: parallax angle= 0.04 Procyon: parallax angle= 0.3
Kappa Ceti: parallax angle= 0.1 GQ Lupi: parallax angle= 0.008
Remember that an inverse proportionality means that as one quantity gets smaller, the other gets larger, and vice versa. That is, the farther away a star is, the larger its distance, and therefore the smaller its parallax angle will be.
So, in order to find the farthest star, look for the smallest parallax angle. Of these four stars, this is star GQ Lupi with parallax angle of 0.008. The closest star is Procyon, because it has the largest parallax angle of 0.3.
Eq. 5.1 can only be used as a distance-measuring tool for objects outside our Solar System, such as other stars. And it isn’t possible to measure parallax for all stars – only the nearby ones in our own Galaxy, where “nearby” in this case means within a few hundred parsecs. At distances larger than that, although the parallax phenomenon still occurs, the angles are so tiny that even the best instruments do not have sufficient angular resolution to detect them, as described in Section 4.3.
124 Stars
5.1.2 Solving parallax problems: absolute method
Given either a distance or a parallax angle, Eq. 5.1 can be used to calculate the other. If the quantity you are given has the appropriate units (parsecs for distance or arcseconds for angle), then you can simply plug in the given value and do the calculation of one divided by that value. Your answer will automat-ically come out in the correct units for the other value. If the number you are given does not have the required units, then you must perform a unit conver-sion before plugging into Eq. 5.1 (for a refresher on unit converconver-sions, take a look at Section 1.1).
Example: How far away is the star Alcor in the previous example?
Alcor has a parallax angle of 0.04, and since arcseconds are the correct units for using the parallax equation, you can plug 0.04 directly into Eq. 5.1. Rewrit-ing the parallax angle as a fraction (0.04 = 4/100) often makes the calculation easier, and in this case it is simple enough that a calculator is not needed:
d= 1 p = 1
0.04 = 1
4 100
=100
4 = 25 = 25 pc.
Since you plugged in a parallax angle in units of arcseconds, the distance you calculated is in units of parsecs automatically. The shortcut employed to get from 4/1001 to 1004 , using the fact that 1 divided by any fraction is simply the inverse of that same fraction, frequently comes in handy when doing parallax problems.
Example: Polaris (the “North star”) is 434 light years away. What is its parallax angle?
This problem requires two steps before you can plug numbers into the parallax equation. First, you should rearrange Eq. 5.1 to solve for the parallax angle p, since p is the quantity you are asked to calculate:
d= 1
p → d × p
1 = 1
p× p 1 = 1,
d× p = 1 → d × p
d = 1 d, p= 1
d. (5.2)
Second, the distance is given in units of light years instead of parsecs, so you must perform a unit conversion. The relevant conversion factor is
5.1 Stellar parallax 125
1 pc↔ 3.26 ly. You can combine the unit conversion step with plugging the value of d into Eq. 5.2:
p= 1
d = 1
434 ly·
3.26 ly 1 pc
= 3.26
434 pc = 3.26
434 arcsec= 0.0075
Using the necessary distance units of parsecs in the denominator guaranteed that the answer for the angle would come out in arcseconds. This angle, about 7.5 thousandths of an arcsecond (or 7.5 milliarcseconds), is readily detectable by modern research telescopes.