Astronomy 300 HOMEWORK #2 Spring ’20 Due date: Thursday, Feb. 13 1. Order-of-magnitude estimate (no calculators!!) On p.8, Croswell gives a value for the speed (in miles per day) with which the Andromeda galaxy (M31) and the Milky Way are approaching each other. (i) Convert this speed to an approximate speed in miles per hour and a speed in km/s and comment. How do these speeds compare with other speeds you know? (ii) At this speed, about how long will it be before the two galaxies collide? Show all your steps in detail. Calculators may now be used J And now pay close attention to significant figures... 2. With an eccentricity of e=0.017, the Earth’s orbit around the Sun is very close to being a circle, but not quite! (i) Combining this eccentricity value with what you know about the Earth’s average distance from the Sun, compute (a) Earth’s perihelion distance (in AU and in km) (b) Earth’s aphelion distance (in AU and in km) (c) The % by which the Earth-Sun distance can be larger or smaller than average. (ii) The Earth moves fastest along its orbit in January and slowest in July. What can you deduce about the time of year that the Earth is closest to the Sun? Explain your reasoning, and state which of Kepler’s laws you made use of. 3. Keplerian orbits and exoplanets In the mid-1990’s, not a single planet was known beyond the Solar System. Since then, thousands of planets have been discovered in orbit around nearby stars in our Galaxy. The star 55 Cancri, in the constellation Cancer, is visible with the unaided eye and is the first star known to have five planets in orbit around it. For the latest information… * Go to the web site exoplanets.org. Click on the “Table” link. This is a table of the known characteristics of confirmed planets outside our solar system. * Notice the following columns in the table (we’ll learn more about these parameters and how they are measured later in the course): column 1 = planet name (e.g., Kepler 107d) column 2 = Msin(i) (this is the planet’s approximate mass in units of Jupiter’s mass) column 3 = semi-major axis (a) in Astronomical Units column 4 = orbital period in days column 5 = orbital eccentricity (e) (unitless) * Type “Cnc” into the filter box at the top (that’s short for “Cancri”) to see the information about the 5 planets (b, c, d, e and f) that orbit the star known as 55 Cancri (ignore “bet Cnc b” which is a planet orbiting a different star known as beta Cancri). Each entry is for one of the five planets. (i) Compare the masses, semi-major axes and eccentricities of the 5 planets in this system to the planets in the Solar System. Write a paragraph comparing the two systems, considering the masses, semi-major axes, and eccentricities in turn. Qualitative comparisons are fine for this part (no calculations needed). (ii) Check Kepler’s 3rd Law for this system by computing P2/a3 for each planet. How well do they all match? (Take care with significant figures!) Does P2/a3 equal 1 for this system as it does for the Solar System? (If not, as we’ll see when we get to Newton’s laws, it means that the mass of the star (55 Cnc) is not the same as the Sun’s mass. Hint: don’t forget to convert the periods to years first! (iii) Compute the periastron, apastron, and semi-minor axis of 55 Cancri f, the planet with the largest orbital eccentricity. Note that the semi-minor axis is given by b = a (1 – e2)1/2 (iv) On a separate sheet of paper, make a sketch of the orbit of 55 Cnc f, clearly indicating where the star is on the plot. Make the sketch one of the following ways: (a) by hand, using tacks and a string or thread (b) by hand, using graph paper and a calculator (c) using a computer Depending on what method you choose, it may help to know that the equation of an ellipse is y = b (1 – x2/a2)1/2 . Choose a scale that will allow you to plot the other planets on your sketch as well (see section (v)). (v) Add the other four planets to your sketch. These planets all have small enough eccentricities that you can assume that the orbits are circular for this sketch. If any of the planets can’t be seen on your sketch, explain in words where they are and why they can’t be seen in your drawing (or your computer-generated plot). (vi) Comment on the effect of 55 Cnc f having such a high eccentricity. Does it run the risk of ever crashing into any of the other planets in the system? Why or why not? .