Astronomy-HoMA Week 1 Homework Exercises V2

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Week 1 Exercises and Activities

February 25, 2014

1 Gas Retention

As we discovered in this weeks lectures, the distance of the planet from its host

star determines its temperature during formation. We also saw that this greatlyaffects the types of gases and ices that the planet will be able to condense andkeep in its atmosphere as it evolves in time.

An important variable that we will learn more about next week is escapevelocity. In short, this is the minimum speed that an object (be it a particle orextended object like a rocket) must have to escape a planets gravitational pull.Because the escape velocity varies with distance to the center of a gravitationalbody (like a star or a planet), the size of the planet will also have an affect onthe escape velocity. You will also see (by interacting with the simulator) thatthe density (and thus mass) of the gravitational body will have an affect on theescape velocity.

If you are having trouble using the simulation there is a help tab in the upperright hand corner that will describe how to use it.

1.1 Link: Gas Retention Simulation

1.2 Goals

Given some combination of a planets parameters (radius, temperature,density), be able to calculate ranges for other parameters so that certaingases are gravitationally bound to the host planet

1.3 Exercises

1. Find the gases that could be gravitationally bound to Titan (a moon ofSaturn)

2. Find the gases that could be gravitationally bound to a planet with aradius of9330 km and a mass of21024 kg and a distance of4.6AU froma sun-like star

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http://astro.unl.edu/naap/atmosphere/animations/gasRetentionPlot.htmlhttp://astro.unl.edu/naap/atmosphere/animations/gasRetentionPlot.html


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3. If a planet is0.3AUfrom a sun-like star with a density of water (1 g/cm3),what is the minimum radius of the planet if it has oxygen and carbon-

dioxide in its atmosphere?

1.4 Solutions

1.We see that Titan just barely has enough mass to hold onto oxygen, andcan also hold onto carbon dioxide and xenon. Every other gas (beginningwith nitrogen) is too light to be gravitationally bound to the moon.

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2.We see that to have a mass of 2 1024 kg we must set the density to

0.6 g/cm3

. To make the planet 4.6AU from a sun-like star we must setthe temperature to 130K. This shows us that the planet will hold allmolecules heavier than (and including) methane, but will not hold hydro-gen or helium. So it can hold methane, ammonia, water, nitrogen, oxygen,carbon dioxide, and xenon.

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3.We see that a planet 0.3AUfrom a sun-like star would have a surface tem-

perature of510K. Since oxygen is harder to hold onto than water, we onlyneed to check that the planet can hold oxygen. If we set the density to1 g/cm3 and adjust the radius until the line passes through our fictitiousplanet we see that the radius must be a minimum of about 8280 km.

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2 Radial Velocity

In this weeks lectures we looked at a number of different ways astronomers canindirectly discover planets around other stars. One of those methods was theradial velocity method, which uses a color shift of an observed star to determineproperties of a planet.

2.1 Link:Radial Velocity Simulation

2.2 Goals

Given a radial velocity graph, be able to find:

1. The eccentricity of its orbit

2. The semi-major axis of the planets orbit

3. The longitude angle (or observing angle)

4. The minimum mass of the planet

2.3 Exercises

1. Given the following observation of a 0.54Msun stars radial velocity, findthe mass, longitude angle, and semi-major axis of the planet if it hasa period of 364 days, an eccentricity of 0.46 and an inclination of 90

2. Given the following observation of a 2Msun stars radial velocity, findthe mass, longitude angle, and semi-major axis of the planet if it has

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http://astro.unl.edu/classaction/animations/extrasolarplanets/radialvelocitysimulator.htmlhttp://astro.unl.edu/classaction/animations/extrasolarplanets/radialvelocitysimulator.html


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a period of 10.1 days, an eccentricity of 0.7 and an inclination of 45.

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2.4 Solutions

Figure 1:

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Figure 2:

1. First we set the parameters we know, namely the mass of the star, the in-clination and the eccentricity. Notice that the only parameter that affectsthe period is the semi-major axis, so we set that next. Now we can adjustthe symmetry of the curve using the longitude. This will only be a roughestimate and should be around 34.7 10 . Finally, the only thing themass of the planet will affect is the magnitude of the radial velocity. Weadjust this to find a mass of about 2.75Mjupiter 0.1Mjupiter. (See fig1)

2. First we set the parameters we know, namely the mass of the star, theinclination, and the eccentricity. Notice that the only parameter that

aff

ects the period is the semi-major axis, so we set that next. Now we canadjust the symmetry of the curve using the longitude. This will only be arough estimate and should be around 180 10 . Finally, the only thingthe mass of the planet will affect is the magnitude of the radial velocity.We adjust this to find a mass of about 4Mjupiter 0.2Mjupiter. (See fig2)

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3 Planetary Transit

Another important planetary detection method we discussed is the planetarytransit, which looks for periodic dips in a stars brightness to determine proper-ties of a planet.

3.1 Link: Planetary Transit Simulation

3.2 Exercises

1. Imagine a system where a hot jupiter with a mass of 10Mjupiter and aradius of2 Rjupiter orbits a host star with mass 2Msun. If it has a circularorbit and a semi-major axis that is the same as the Earths, what is themaximum amount the system can be tilted from 90 for us to be able todetect a planetary transit? What is the length of this planets year? How

long does the transit last?

2. For the system described in problem 1, what is the maximum amount thesystem can be tilted from 90 if instead the planet has the same mass andradius as Earth (you can look these up, for example: go to google andsearch for mass of earth / mass of jupiter)?

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http://astro.unl.edu/naap/esp/animations/transitSimulator.htmlhttp://astro.unl.edu/naap/esp/animations/transitSimulator.html


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3.3 Solutions

Figure 3:

Figure 4:

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Figure 5:

1. First we enter the information for the planet and star properties. Next westart to lower the inclination of the orbit. We see that if we go more thana fraction of a degree below 89.5 the orbit of the planet never crosses thestar, so the maximum the orbit can be tilted is 0.5. This planet has a258 day year, and a transit that lasts 68.4 minutes. (see figure 3)

2. We first look up the mass and radius of the Earth in units of Jupiters massand radius. We see that MEarth/MJupiter =.003146and REarth/RJupiter =.08921 (see figure4). Next we adjust the inclination, just like we did forproblem 1 and see that the maximum inclination is about 89.55. Noticethat the length of the year is the same as for a larger planet (this is be-cause the orbit only depends on the mass of and distance to the star andnot the mass of the planet).

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4 Habitable Zones

Just by looking in our own solar system we see that not every planet (and moon)is suitable for life. Recall this weeks lectures on the habitable zone, a very roughestimate of the range of distances from a star that liquid water can exist on aplanet (as opposed to all of a planets water either freezing or being vaporized).

Next week we will learn about stellar modeling and how a star evolves intime. For now, all you need to know is that the size and temperature of astar varies in time, both of which affect the amount of energy a planet receivesand impacts its habitable zone. In this simulation we investigate how stellarevolution affects the habitable zone of planets, telling us how long a planet hasto evolve.

4.1 Link: Habitable Zone Simulator

4.2 Goals

For a given star, understand how its habitable zone changes in time

4.3 Exercises

1. Lets first begin with the Earth. Since the Earth is believed to be about4.54 billion years old. At what time was the Earth first in the habitablezone? At what time will the Earth leave the habitable zone?

2. Assuming that life first appeared when Earth entered the habitable zone,and exists until Earth leaves the habitable zone, how much total time willlife exist on Earth? How much time do we have left before the planet

becomes uninhabitable? Will all the water freeze or boil off

?3. Now choose the star Gliese 581. There are a few things to notice before

we answer this question. First we see that this star will live a lot longerthan the sun. We also see that the planet Gliese 581 d is tidally locked tothe star, meaning the same side is always facing the star, baking one sideand freezing the other. Lets assume that life is able to form on a tidallylocked planet as long as its center is located in the habitable zone . Atwhat time does planet d first enter the habitable zone? At what time doesplanet d leave the habitable zone?

4. How long does life have to arise on this planet? Assuming being in thehabitable zone is the only important factor on the possibility of life on aplanet (it isnt, this is an oversimplification), is life more likely to form on

a low mass star or a sun like star?

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http://astro.unl.edu/naap/habitablezones/animations/stellarHabitableZone.htmlhttp://astro.unl.edu/naap/habitablezones/animations/stellarHabitableZone.html


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4.4 Solutions

Figure 6:

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Figure 7:

1. We see that at time t=0 the Earth is already in the middle of the habitablezone (see figure6). We see that as we increase the time, the habitable zonemoves outward and according to this model the last time the earth is inthe habitable zone is around 5.42 billion years.

2. This means that life will have approximately 5.42 billion years to form onEarth. It also means we only have about 0.8 billion years left before ourplanet is no longer habitable.

3. The planet first enters the habitable zone around 250 billion years (18times longer than the current age of the universe). It leaves the habitablezone around 393 billion years. (see figure7)

4. This gives the planet about 143 billion years for life to arise! Assumingbeing in the habitable zone is the only requirement for life, this makes itMUCH more likely life will form around a low mass star than a sun likestar.

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Astronomy-HoMA Week 1 Homework Exercises V2

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