(a) Using your answers to Problems 4-9 and 4-10 and assuming that all energy transport inside Ju

(a) Using your answers to Problems 4-9 and 4-10 and assuming...

Key Concepts

Thermal Conduction

Thermal conduction is the process by which heat is transferred through a material due to the temperature gradient. In the context of planetary interiors, it is modeled using Fourier's law, which relates the heat flux to the temperature gradient by the material's thermal conductivity. This concept is foundational for estimating how energy moves through dense materials such as those found in Jupiter's interior.

Thermal Conductivity

Thermal conductivity is a property of materials that indicates their ability to conduct heat. In a planetary context, values for thermal conductivity are crucial for determining how efficiently energy is transferred by conduction inside planets. High conductivity implies that a material can readily transport internal heat, affecting the estimated temperature gradients and overall thermal state of planetary interiors.

Energy Transport in Planetary Interiors

Energy transport in planetary interiors can occur via conduction, convection, or radiation, with each mechanism having different implications for internal temperature gradients. For gas giants like Jupiter, assumptions about energy transport methods significantly influence temperature estimates. Typically, convection is the dominant mode, but exploring conduction helps highlight the impact of different physical processes on planetary structure and evolution.

Model Assumptions and Their Validity

When modeling planetary interiors, assumptions regarding the dominant heat transport mechanisms are crucial. In the given problem, it is assumed that conduction is solely responsible for energy transport. However, if the resulting interior temperature estimates are implausibly high compared to known values, it indicates that this assumption may be incorrect. This highlights the importance of validating model assumptions by comparing results with independent constraints from observational data or more comprehensive theoretical models.

Transcript

00:01 So here we need to find the root mean squared velocity of hydrogen molecules on jupiter on earth, find the escape velocity for both of these planets, and then see if hydrogen can make it out of both of these planets.

00:17 Part d asks us to find the escape velocity and the root mean squared velocity on an asteroid series, and see if an oxygen atmosphere can be established.

00:31 So we should first write down all of our givens, and there are a few.

00:36 So we have the temperature of jupiter being 140 kelvin.

00:41 And then the temperature of earth's troposphere at 20 kilometers.

00:47 So we're just going to say t sub e, 220 kelvin.

00:53 And then we can say y equals 20 kilometers, because this is the temperature.

00:58 In the troposphere.

01:00 Then we have the molar mass of hydrogen being of 0 .002 kilograms per mole.

01:11 We have the mass of earth equaling 5 .97 times 10 to the 24th kilograms.

01:22 We have the radius of earth equaling 6 .38 times 10 to the 6 .3, then we have the mass of jupiter equaling 1 .90 times 10 to the 27th.

01:39 These are all given and they're tabulated as well.

01:44 And then we have the radius of jupiter equaling 6 .91 times 10 to the 7 meters.

01:58 So at this point we can then find all of our, all of the velocities that we must find.

02:05 So for part a, we're going to be on jupiter.

02:09 So let's find the root mean squared velocity of the hydrogen molecules on jupiter.

02:16 So that would be equal to 3rt over the molar mass of hydrogen.

02:22 And this is going to be equal to 3 times 8 .314 times the temperature of 140, all divided by the molar mass of hydrogen.

02:39 So that'll be .00202.

02:43 And this is going to be equal to 1 ,314 .78 meters per second.

02:52 The escape velocity can be found by this formula.

02:57 So this would be the square roots of three, rather two.

03:06 My apologies, that would be two times the universal gravitational constant, times the mass of jupiter divided by the radius of jupiter.

03:20 And then we can solve directly and say 2 times 6 .67 times 10 to the negative 11th.

03:34 And then the mass of jupiter being 1 .9 times 10 to the 27th.

03:41 I apologize.

03:42 All divided by the radius of jupiter 6 .91 times 10 to the seventh.

03:55 I apologize for it being so cramped.

03:59 And then we find that this is going to be equal to 6 .06 times 10 to the fourth meters per second.

04:12 So at this point we can say, okay, let's get a new workbook here and do b.

04:19 Rather let's before we go that we need to compare these so we can say that vrms is going to be equal to 0 .221 the escape so just keep that in mind the root mean squared velocity of the hydrogen molecules is 22 .1 % of the escape velocity on jupiter and then for b we can have earth and then we find all these for earth so v rms on earth it would be equal to again three rtm and we can substitute directly so it would be three 8 .314 times 220 kelvin all divided by the molar mass so that would be 0 .00202 and then this would equal 1 ,648 .17 meters per second and then at this point, we can say, okay, the escape velocity is going to be, again, 2g times the mass of earth divided by the radius of earth.

05:38 This is going to be equal to 2, 6 .67 times 10 to the negative 11th, times the mass of earth, which would be, again, 5 .97 times 10 to the 24th kilograms, all divided by the radius, which would be a 6 .97.

06:03 0 .38 times 10 to the 6 meters and this is going to be equal to 1 .12 times 10 to the 4 meters per second.

06:26 At this point we can then again compare it so vrms would be equal to 0 .147 times v.

06:41 Escape.

06:43 So the root means squared velocity of these hydrogen molecules on earth would be 14 .7 % of the escape velocity on earth.

06:59 And then c is asking us, why is jupiter's atmosphere's 89 % hydrogen and earth's atmosphere only has, you know, trace elements, trace percentages of hydrogen? well, the rms speed, the rms of the rms speed of hydrogen molecules is greater on earth than on, so the vms speed of hydrogen molecules on earth is greater than the vrms of hydrogen molecules on jupiter.

07:43 In addition, the escape velocity on earth is less than the escape velocity on jupiter.

08:06 So because of this, that means that, well, of course, it's more likely for h2 molecules to escape earth, to escape earth than to escape jupiter.

08:37 And that's due to this...

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