Homework4 _2023

PHSC 10800: Earth as a Planet Homework #4 Assigned October 27, 2023 Due November 3, 2023 1 ) Recent Planetary Science (10 pts) Last year, the DART (Double Asteroid Redirection Test) Mission successfully demonstrated that we have the technology to change the orbit of a potentially hazardous asteroid. It did this by crashing a projectile into the asteroid Dimorphos, which is a satellite of the larger asteroid, Didymos. a) Prior to the DART mission, the orbital period of Dimorphos was approximately 11 hours and 55 minutes, with a semi-major axis about Didymos of nearly 1.2 km. What is the mass of Didymos? b) After impact, the orbital period of Dimorphos changed to 11 hours and 23 minutes. How much did the semi-major axis of the asteroid change about Didymos as a result of the mission? Give your answer in meters and as a fraction of the original semi- major axis. 2) Moon’s orbital evolution (10 pts) As discussed in class, due to tides, the moon is receding from the Earth at a rate of ~4 cm/year. Let’s assume that rate has been constant throughout time (it wasn’t, but we can use it to illustrate some key points). The moon’s current semi-major axis is 384,400 km. a) If the moon formed 4.5 billion years ago and has been receding from the Earth ever since, what was its original semi-major axis? What was its original orbital period? b) What would the apparent size of the Moon have been in the sky as viewed from Earth? That is, in Hmwk 2, you were told the diameter of the Moon spans about 0.5 o when viewed from Earth today. What would it have been when the Moon first formed? 3 ) Shifting Moonrise (10 pts) The timing of Moonrise is later every day. Let’s understand why: a) Moonrise occurs when you, as you sit on the rotating Earth, turn to the point where you can just see the Moon appear over the horizon. Draw a picture to illustrate what this would look like. Draw the Earth, Moon, and you standing on the Earth to indicate this scenario. (You’ve seen my artistic skills in class, I’m not asking you to do anything better than that. Just label/make clear what is shown). Take this as Day 1. In 24 hours, you would return back to that exact point on the Earth. However, the Moon will also have moved forward in its orbit (it orbits the Earth in the same direction that the Earth is spinning). How far along in its orbit does the Moon move?

That is, assume the Moon is orbiting in a circular orbit; in going from Day 1 to Day 2, how many degrees does thee Moon move forward? Indicate (very roughly) via a drawing. b) Because the Moon advanced in its orbit, you now must wait longer to see the Moonrise. Sketch this. If you must wait for the Earth to rotate the same number of degrees as the moon moves along in its orbit in part a to see this, how much longer do you have to wait? Answer in minutes. [ NOTE: This exercise here is relevant to your Moon Journal. The Moon’s orbit is not perfectly circular, but elliptical. As a hint for your Moon Journal, think about how the thought exercise in this problem would be different if you allowed for the Moon to follow a non-circular orbit, keeping in mind what you know about the movement of an object as given by Kepler’s Laws. You don’t have to answer anything here—this is meant to nudge you in the right direction as you think about your Moon Journal. ] 4) Lunar Eclipses (10 pts) Further below is a sketch of what the Earth’s Umbra and Penumbra look like and what determines their sizes as presented in lecture. For simplicity’s sake, let’s assume the Moon orbits the Earth in the same plane as the Earth’s orbit (there is no inclination). For clarity, the Penumbra is all the shadow that is not in the conical (triangle in picture below) Umbra. a) Calculate how far the Earth’s Umbra extends beyond the planet. That is, note the edge of the Umbra extends along lines that are tangent to the Earth and to the Sun. Two similar triangles can be drawn from the far point (terminus) of the Umbra: one with a base of the diameter of the Earth, and one with the base the diameter of the Sun. Give your answer in km and in terms of the Moon’s semi-major axis (384,400 km). b) If the Earth and Moon moved to Venus’s orbit (semi-major axis of 0.7 au), with everything else remaining the same, would the Moon still experience total Lunar eclipses if its distance from Earth remained the same? 5) Solar Eclipses (10 pts) A Total Solar Eclipse occurs when the Moon blocks out the light of the Sun. In other words, this would be when the Moon’s Penumbra intersects the surface of the Earth. Draw the above picture, but with the Moon and Earth swapped (you can take the Moon as being located exactly 1 au from the Sun, with the Earth beyond it)? a) How far would the Moon’s Umbra extend? Give your answer in km. b) Remember that the Moon has an eccentricity of 0.0549. How does the length of the Moon’s umbra compare to the Earth-Moon distance when the Moon is at perigee? Apogee? What does this mean about the types of solar eclipse that would be seen when the Moon is at these locations?

DISCUSSION BOARDS 1) Look back at the memes you and your group produced. Pick the top 3 to share with the class. They can be attributed to you when you share or you can be anonymous. 2) Find a light source and a roughly spherical object to be illuminated by that light source to recreate the phases of the Moon. Take pictures to show that you can create 3 of the possible phases of Full, Gibbous, Quarter, Crescent or New Moon (identify the phases when you turn them in). If you are unable to take pictures, sketch/draw/etc what you have done to produce the 3 di ff erent phases. Describe your set up, and the orientations that lead to the phases you produced. As a discussion group, provide as many variations in the phases as possible. Which do you think would best illustrate the concepts of “New Moon,” “Quarter Moon,” “Crescent Moon,” “Gibbous Moon,” and “Full Moon” to an elementary school audience? What about the di ff erences in “Waxing” or “Waning” phases. Note: this requires you to look at other people’s postings and respond—waiting to the last minute to post is not helpful to you or your peers. Relevant Numbers 1 AU = 150,000,000 km = 1.5x10 8 km Eccentricity of Earth’s Orbit: 0.0167 Radius of Earth: 6371 km Mass of Earth: 5.96x10 24 kg Radius of the Moon: 1737 km Mass of Moon: 7.34x10 22 kg Radius of Mars: 3390 km Mass of Mars: 6.4x10 23 kg Radius of the Sun: R ⦿ =696,300 km Mass of the Sun: M ⦿ =2x10 30 kg Gravitational Constant: G=6.67x10 -11 m 3 /(kg s 2 ) Speed of light: c=3x10 8 m/s Relevant Equations Kepler’s 3rd Law: P 2 = (4 π 2 /GM center ) a 3 Here, M center is the mass of the central body around which things orbit, such as the Sun. This is true if the orbiting body has a small mass compared to the central body . Newton’s Law of Universal Gravitation: F = GMm/d 2 Here, M is the mass of one body, m the mass of another, G the universal Gravitational Constant, and d the distance between the centers of the two objects. λ max = 0.0029 m K T T F = 9 5 T C + 32 T K = T C + 273