HW5 Q2, Q3, Q5 (William Matzko)
2. We know the phase amplitude is ε = A*g*(R/r)^2, where A is the geometric albedo, R is the radius of the planet, r = a(1-e^2)/(1+ecos(f)), and g = 10^(-0.4*m). Here, m is just
m = 0.09(α/100) + 2.39(α/100)^2 - 0.65(α/100), where α is in degrees (the 100's in the denominator are likewise in degrees). We are working with a Venus analog, so we can assume, in this part, that e = 0, so that r = a. The phase amplitude will be maximum when α = 0. So, our maximum phase amplitude is then ε = (0.67) (1)(6050km/108E6km)^2, where I have plugged in the geometric albedo of Venus, the radius of Venus, and Venus' semimajor axis. This yeilds ε = 2E-9.
Now, we are to assume i = 40 degrees and w = 270 degrees. The effect of w is to orient the orbit such that we have a maximum phase angle at f = 0 . We know the relation between α and these orbital elements: cos(α) = -sin(i)sin(w+f). Plugging in the numbers and solving for α yields α = 50 degrees. We can then compute m using the same equation as above. This yields m = 0.3175. Computing g again using the same equation as above, we find that g = 0.746. Venus' albedo remains unchanged, and with f = 0, we have r = a(1-e). Thus, ε = (0.67)(0.746)(R/(a(1-e)))^2. Rearranging and simplifying gives us (1-e)^2 = (0.5) (R/a)^2(1/ε) = 0.7854. And so e = 0.114 is the eccentricity that will reproduce the maximum phase amplitude as calculated above.
3. (a) One early surprising feature was Super Earths. These are planets between the size of Earth and the size of Neptune. Kepler revealed that these types of planets are incredibly common, and yet there are none of these kinds of planets in our Solar System, hence the surprise. Another surprise was the existence of Hot Jupiters. These planets are, obviously, about the size of Jupiter and orbit incredibly close to their host star. These were surprising because, again, we have nothing like these in our Solar System. Yet, Kepler revealed that Hot Jupiters are quite common.
(b) Kepler found a relation between planet size and occurrence. Namely, Kepler found that nature tends to create 3 relatively distinct classes of planets: Hot Jupiters, Cold Jupiters, and Super Earths. Super Earths and Hot Jupiters are very common, while Cold Jupiters aren't. As a general trend, "smaller" (~Neptune size and less) planets are significantly more common than ~Jupiter size planets.
(c) The Habitable Zone is defined as the range of distances from a planet's host star where liquid water could potentially exist on the planet's surface.
5. (a) Two properties of the initial disk that determine what kinds of planets will form are temperature and mass. If the disk has more mass, there is more matter to make planets and thus there can be bigger planets (and in bigger abundance) than other solar systems. The average temperature of the disk could also effect how gasseous planets and ice giants form.
(b) The first basic type of planetary migration is called "Type 1." This is when a planet, typically less than 10 Earth masses, moves through the disk in such a way that the spiral perturbations its movement causes are linear. In general, the structure of the disk itself is not significantly disturbed by this type of migration. The second type is "Type II." Here, angular momentum is exchanged between the planet and disk via wave excitation and shock dissipation. The significant difference between Type I and Type II is that Type II causes significant disruption to the disk. As the planet moves through the disk, it traces out a "gap." Type III is the last type of migration, and is similar to Type I. The main difference is that Type III migration is much faster than Type I. Also, "co-rotational resonance" perturbs the disk here instead of waves.
m = 0.09(α/100) + 2.39(α/100)^2 - 0.65(α/100), where α is in degrees (the 100's in the denominator are likewise in degrees). We are working with a Venus analog, so we can assume, in this part, that e = 0, so that r = a. The phase amplitude will be maximum when α = 0. So, our maximum phase amplitude is then ε = (0.67) (1)(6050km/108E6km)^2, where I have plugged in the geometric albedo of Venus, the radius of Venus, and Venus' semimajor axis. This yeilds ε = 2E-9.
Now, we are to assume i = 40 degrees and w = 270 degrees. The effect of w is to orient the orbit such that we have a maximum phase angle at f = 0 . We know the relation between α and these orbital elements: cos(α) = -sin(i)sin(w+f). Plugging in the numbers and solving for α yields α = 50 degrees. We can then compute m using the same equation as above. This yields m = 0.3175. Computing g again using the same equation as above, we find that g = 0.746. Venus' albedo remains unchanged, and with f = 0, we have r = a(1-e). Thus, ε = (0.67)(0.746)(R/(a(1-e)))^2. Rearranging and simplifying gives us (1-e)^2 = (0.5) (R/a)^2(1/ε) = 0.7854. And so e = 0.114 is the eccentricity that will reproduce the maximum phase amplitude as calculated above.
3. (a) One early surprising feature was Super Earths. These are planets between the size of Earth and the size of Neptune. Kepler revealed that these types of planets are incredibly common, and yet there are none of these kinds of planets in our Solar System, hence the surprise. Another surprise was the existence of Hot Jupiters. These planets are, obviously, about the size of Jupiter and orbit incredibly close to their host star. These were surprising because, again, we have nothing like these in our Solar System. Yet, Kepler revealed that Hot Jupiters are quite common.
(b) Kepler found a relation between planet size and occurrence. Namely, Kepler found that nature tends to create 3 relatively distinct classes of planets: Hot Jupiters, Cold Jupiters, and Super Earths. Super Earths and Hot Jupiters are very common, while Cold Jupiters aren't. As a general trend, "smaller" (~Neptune size and less) planets are significantly more common than ~Jupiter size planets.
(c) The Habitable Zone is defined as the range of distances from a planet's host star where liquid water could potentially exist on the planet's surface.
5. (a) Two properties of the initial disk that determine what kinds of planets will form are temperature and mass. If the disk has more mass, there is more matter to make planets and thus there can be bigger planets (and in bigger abundance) than other solar systems. The average temperature of the disk could also effect how gasseous planets and ice giants form.
(b) The first basic type of planetary migration is called "Type 1." This is when a planet, typically less than 10 Earth masses, moves through the disk in such a way that the spiral perturbations its movement causes are linear. In general, the structure of the disk itself is not significantly disturbed by this type of migration. The second type is "Type II." Here, angular momentum is exchanged between the planet and disk via wave excitation and shock dissipation. The significant difference between Type I and Type II is that Type II causes significant disruption to the disk. As the planet moves through the disk, it traces out a "gap." Type III is the last type of migration, and is similar to Type I. The main difference is that Type III migration is much faster than Type I. Also, "co-rotational resonance" perturbs the disk here instead of waves.
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