Questions about a particular planet

[Apollo7]

I have created a solar system and it is inhabited by a planet with some interesting characteristics. I'd like to ask the brain trust here to explain to me a certain situation I've run into.

This particular planet is a frigid 45K, obviously not a summer vacation spot. It has 2.16 Earth Masses and a radius of 13779.84 Km. It's density is just 2.46 gm/cm^3 and it rotates once on its axis every 84.24 hours.

The question is thus: I've found that this planet has an escape velocity of about 16.16 Km/s and an acceleration due to gravity of 9.47 m/s at the surface. The interesting thing here is the planets acceleration due to gravity is in fact smaller than the figure for the Earth, while its escape velocity is decidedly higher. I'm wondering how exactly this can be? In every other case in my system both the EV and Acel are in agreement, either being smaller or larger than the values for Earth. What about this particular planet has broken the mold?

Cheers.

[JrzyCrim]

Based on your planet's characteristics, here is what I came up with:

I am using the M K S system (Meter, Kilogram, second)

The gravitational constant:

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G=6.67259e-11 M^3/Kg*S^2

The formula for calculating the acceleration of gravity @ the surface is:

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Ag=GM/R^2 (G=Gravitational constant, M=planet's mass, R= Radius)

Escape Velocity:

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Ve=(2GM/R)^.5

The mass and radius of the Earth:

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M=5.974e24 Kg  R=6.371e6 m

The mass and radius of your planet:

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M=1.29e25 Kg  R=1.3779e7 m

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The Ag of Earth= 9.82 m/s^2.
The Ve of Earth= 11.2 km/s

Here are my results for your planet.

The Acceleration of gravity:

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Ag=6.67259e-11 * 1.29e25 / 1.3779e7^2 = 4.53 m/s^2

The escape velocity:

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Ve=(2*6.67259e-11*1.29e25 / 1.3779e7)^.5 = 11.2 km/s

Escape velocity and gravational acceleration do not vary by the same amount with changing mass and radius. As you can see by the above equations, Escape velocity increases by the square root of the mass (double the mass, Ve increases by 1.414 times) and decreases by the inverse of the square root of the radius (double the radius, Ve increases .707 times or decreases 1.414 times -- same thing)

The interesting thing is that if you increase both the mass and the radius of a given planet by the same proportion, the escape velocity will remain the same. The mass of your planet is 2.16 times that of Earth, the radius is also 2.16 times that of Earth so the escape velocity remains the same.

Gravitational acceleration, on the other hand, varies proportionately with mass (double mass, Ag will double) and Ag varies by the inverse square of the radius (double the radius, Ag will be 1/4 as great, half the radius Ag will be 4 times greater)


These calculations are based on the given mass and radius of your planet.
I noticed a discrepancy between the mass/radius values and the value for the given density.

Based on a Mass of 1.29e25 kg and and a radius of 1.3779e7 m, the density should be 1177 kg/m^3 (1.177 g/cm^3). A little less than half of your given value. This is most likely where the discrepancies lie.

I calculated the density by Mass/Volume, Volume being 4/3*pi*R^3.

Hope this helps.

[Apollo7]

Thank you, very informative, however the mass of my planet (in this case) is 4.510492Me or about 2.69*10^25 Kg. Again this is not to say you're wrong only you took the (apparently) wrong value for mass. The R = 2.160 and the M = 4.510.

Thanks though!

[JrzyCrim]

This particular planet is a frigid 45K, obviously not a summer vacation spot. It has 2.16 Earth Masses and a radius of 13779.84 Km. It's density is just 2.46 gm/cm^3 and it rotates once on its axis every 84.24 hours.

Cheers.

I'm glad you found my information useful. However, as quoted above, Your planet is 2.16 Earth masses.

I guess this is a typo on your part. So with the new mass value of 2.694e25 kg, the values are:

Ag=9.467 m/s^2
Ve=16.16 km/s

In agreement with your original values. Doh!

Of course, my calculated value for density was different also because of the incorrect mass value I was using. Doh! Doh!


Look at all that work you put me through. haha. Actually, it was very informative for me as well. I never really examined the escape velocity equation that closely and I learned something about it. I love celestial mechanics.

Regards,

[Apollo7]

Well you know theres no satisfaction like solving a problem you created yourself. Heh.

For me I was just opining on an interesting quirk, I guess with how my mind works I found it intriguing that a planet could have an escape velocity in excess of Earth's, while retaining an acceleration due to gravity which is in fact smaller than the Earthly value.

Of course in my modeling scenarios with the moons of these planets I've found that overall density can be a big factor when dealing with a planets overall characteristics. I have one moon which has a radius of 5688.93 Km and a Mass of 1.4948*10^24. Making its density just 1.938 gm/cm^3. Its interesting because despite having a radius of 90% that of Earth. It's EV is a shade under 6 Kmps and its accelG at just 3 m/s.

I'm thinking I might downsize my system anyway. At present it has 4 gas giants, with a total mass of over 1400Me. Of course by the same token Gas Giants allow for moons and other interesting situations to be explored. My earthlike world in this scenario orbits around a 874 Earth Mass Giant. Of course I placed the planet 1,653,954Km away from the Giant, giving it a period of 197.74 Hours, with synchronus rotation. The Terran world itself has a radius of 6834.96Km and a Mass of 6.93*10^24Kg. The density of 5.185gm/cm^3 means this planet is composed of slightly lighter mateiral than our own Earth.

One thing I'm intersted in finding out now is the power of the magnetic fields of these giant worlds I'm creating, I hear the strength of the field has profound effects on nearby worlds, which was partly why I placed my Earth-like moon so far away from its parent giant. Incidentally the giant is so large that even at the distance quoted above it still appears with an angular size of 6.69 degrees. Of course with a radius of 96571.32Km this giant is larger than some very low mass stars. Well anyway thanks for the replies.

Cheers.

[Evil Dr Ganymede]

I'd guess your moon will be beyond the nasty parts of the jovian's magnetic field at that distance. It ought to get some continuous aurorae though, since I'd imagine the moon has a magnetic field of its own

That radius sounds kinda big though... it's a 2.75 Jupiter Mass jovian, right? You'd only get it that large if it's a very young system (about 50-70 million years old) - they contract over time. I got that info from this paper

[Apollo7]

that is a good point Dr. Ganymede. I know there is an upper limit on the radius of a jovian world before it shrinks and becomes more dense. However I do not truly know how to model this or what the actual limit is so I will check out your link. I know red dwarfs that are low-mass and about the size of jupiter can have a density of something like 200 gm/cm^3 which is damn high. I guess I'll have to check out some background and see what the constraints are.

Cheers.

[Evil Dr Ganymede]

You be needing to check out the graphs on pages 722 and 724 for the useful stuff. The paper talks about jovians from about Saturn's size to Brown Dwarfs of 80 Jupiter Masses. It's obviously a bit technical but there's some bloody useful stuff in there.

Not least of which is the description of the physical appearance of five proposed classes of superjovian (depending on temperature) on page 756. If that stuff got into Celestia it'd REALLY rock.

[Cormoran]

Yep, that's a helluva paper, with some wonderful stuff in it (as the good, I mean bad Doctor says, especially page 756)

Highly reflective Gas Giants, huh? maybe get something with some specular highlights? Damned if I know. The technical stuff gave me a migraine. I think I'll go back to pounding the Earth with asteroids...

Cheers,

Cormoran