7.14. We seek to know how far away we would have to place a dime (actual size s = 1.8 cm = 0.018 m) for it to have an angular diameter of 0.05 arcsecond. We begin by converting this angular diameter from arcseconds to degrees:

We now use this value as the angular diameter a in the angular separation formula and solve for the distance d:

A dime has an angular size of 0.05 arcsecond when seen at a distance of about 74,000 meters, or 74 km. Thus, the Hubble Space Telescope could resolve the disk of a dime (i.e., know that it is a disk and not a point) at a distance of 74 km.

7.16. We use the formula from Mathematical Insight 7.2 to find the diffraction limit of the human eye, but instead of having a telescope diameter we use the lens diameter of the eye, given as 0.8 cm:

(Note that the actual angular resolution of the human eye, about 1 arcminute, is not as good as the diffraction limit, which is a theoretical limit for a perfect optical system.) For a 10-meter telescope, the diffraction limit resolution is:

The diffraction limit of the 10-meter telescope is smaller than the diffraction limit of the human eye by a factor of about 16/0.0125 = 1,280, or close to 1,300.

8.10. Ices would have condensed in the inner solar system, significantly increasing the size and mass (or possibly number) of terrestrial planets. Water and other hydrogen compounds would be much more abundant.

8.11. Without nebular capture, jovian planets would not accumulate substantial amounts of material into the planets themselves or into the disks surrounding them. The jovian planets would consist of icy cores without the envelopes of light gases (H, He). No satellites would form, as no disk would have formed.

8.18. a. The planet around 51 Pegasi has an orbital period of 4.23 days. Because the star has about the same mass as the Sun, we can find the planet's distance from its star by using Kepler's third law in its original form. But first we must convert the period of 4.23 days into units of years:

Now we can use Kepler's third law to find the planet's distance in AU:

The planet around 51 Pegasi orbits with an average distance (semimajor axis) of 0.051 AU - much closer to its star than Mercury is to our Sun.

b. At 0.6 times the mass of Jupiter, the planet in 51 Pegasi seems likely to be a jovian planet. In the nebular theory, jovian planets form outside the "frost line," which in our solar system is outside the orbit of Mars (1.5 AU). Thus, it is surprising to find a jovian planet that is located closer to its star than Mercury is to our Sun.

c. There are many possible explanations for why a large planet might be located so close to 51 Pegasi. The frost line might have been closer in the 51 Peg system; this might have been the case if, for example, the nebula cooled faster in this system than in our own solar system. It is also conceivable that the 51 Peg nebula was much denser and allowed larger planets to form; in this case, the planet might actually be a very large terrestrial planet. Most planetary scientists favor a third possible explanation: that the 51 Peg planet formed farther out but migrated closer in due to interactions with the solar nebula (e.g., drag).

9.8. a. The spacecraft should include a magnetic field detector, because the size and rapid rotation of the planet would be expected to cause a magnetic field if the core is metallic. The spacecraft should also measure the gravitational pull of the planet on the spacecraft (giving the mass) and the size of the planet; together these quantities provide the planet's density.

b. Without radioactive elements, the planet's interior should have cooled enough to end volcanic and tectonic activity. Images of the surface should check whether such features are present or whether they have been covered in craters.

c. Given the planet's size and rotation rate, erosional features should be present if there is an atmosphere.

9.9. a. Radioactive dating is usually considered more reliable than measuring crater abundances partly because it is much more precise. In addition, cratering is somewhat random and more likely to be misleading. Moreover, the precise time at which the early bombardment ended is not well known.

b. Crater abundances are easier to measure on other planets, because it is much cheaper to take photographs than to land on the surface and analyze rocks for radioactivity - either with an intelligent robot or by returning the sample to Earth.

10.8. a. Venus's cloudiness causes it to reflect so much sunlight that it actually absorbs less than the Earth.

b. In the absence of clouds, Venus's dark surface would lead to the absorption of much more sunlight and higher temperatures. Without performing a calculation, it's debatable whether Venus would be warmer with the low albedo or the greenhouse effect. (A calculation would show that the low albedo would not warm the planet as much as the current greenhouse effect.)

c. The clouds contain sulfuric acid probably derived from volcanic outgassing. If outgassing ceased, the amount of sulfur compounds - and therefore clouds - would probably decrease.

10.9. a. With no greenhouse gases, the troposphere would not be warmer at the bottom.

b. Without UV light, no stratosphere would form.

c. With greater X-ray output, the thermosphere and exosphere would be warmer.

10.11. In the daytime, the warmer air over land rises and draws air from the sea to replace it. This leads to winds from sea to shore. At night, the air over the sea is warmer, so the circulation is reversed. The circulation pattern resembles Hadley circulation, with the warmer region (sea or shore) corresponding to Earth's equatorial region.
This explains the smog in cities like Los Angeles; since the wind is coming from the ocean during the day, the air pollution from cars can't easily leave the valley the city is built in.