Question about Mars core

[supermanpj4444]

Would adding more mass to the planet to increase the gravity have any effect on the core? Would the increased gravity speed the rotation of the core and create a magnetic field?

[ajb]

If you want to think about how adding or taking away mass effects the rotation speed of planet you need to think about angular momentum and its conservation. Look up angular momentum and let us know if you think adding mass, but not changing the radius of the planet significantly will speed it up or slow it down?

[csmyth3025]

Would adding more mass to the planet to increase the gravity have any effect on the core? Would the increased gravity speed the rotation of the core and create a magnetic field?

As ajb points out, if you're thinking in terms of hypothetically adding or subtracting mass to a (spherical) rotating body, the equations for determining angular momentum will give you the the answer by assuming that the the mass of the body is uniformly distributed throughout and the radius of the body remains unchanged (it just becomes uniformly more dense).

 

The formula for angular momentum for a rigid (that is, solid) body that's rotating around a fixed axis is:

 

For an object with a fixed mass that is rotating about a fixed symmetry axis, the angular momentum is expressed as the product of the moment of inertia of the object and its angular velocity vector:

 

where I is the moment of inertia of the object (in general, a tensor quantity), and ω is the angular velocity.

 

(ref. http://en.wikipedia....ntum#Definition )

 

So far, so good. Now the angular velocity (ω) is usually given in radians per second, although it may be measured in other units such as degrees per second, revolutions per second, revolutions per minute, degrees per hour, etc. This seems simple enough. Since there are 2*pi radians in a complete circle, one revolution per second would equal 2*pi radians per second (6.28 radians per second).

 

Now we come to the moment of inertia part. This is a bit more complicated, but Wikipedia has simplified it for us:

 

For a Ball (solid) of radius r and mass m:

 

 

 

(ref. http://en.wikipedia....ents_of_inertia )

 

This is where things get sticky (for me, at least). We have the formula for angular momentum L, which is (). We know that ω (angular velocity) is in terms of radians per second. And we know that .

 

For the sake of simplicity, lets say we have a 1000 kg solid ball that's 1 meter in radius rotating at 60 rpm (6.28 radians per second). The angular momentum would then be:

 

L = {(2*1000 kg)*(1 meter)^2)/5}*(6.28/s) = (400 kg*m^2)(6.28/s) = 2512 kg*m^2/s

 

The angular momentum of this object has a direction (the direction of its spin). I believe that makes it a vector quantity, although it may be a tensor (I really don't know what the difference between the two is).

 

All of this is well and good (assuming I've done it correctly). Unfortunately, I'm stuck at this point. My guess is that just adding mass would actually slow down the rotation of the core in order for the result of the above equation (the angular momentum) to remain the same. If, however, adding mass causes the object to become more compressed (smaller radius), then it could conceivably cause the object to speed up.

 

I don't think any of this will cause the core to "create" a magnetic field. If it produced a magnetic field before you added the additional mass, the magnetic field might become stronger or weaker - depending on whether the object ends up spinning faster or slower. As far as I know planetary magnetic fields are caused by conductive fluid motions in the interior of a planet - they're source is dynamically driven, as opposed to the static magnetic fields we normally associate with bar magnets.

 

As always, I may have done this entirely wrong, so corrections are welcome.

 

Chris

 

Edited to correct spelling errors

Edited August 28, 2011 by csmyth3025

[ajb]

My guess is that just adding mass would actually slow down the rotation of the core in order for the result of the above equation (the angular momentum) to remain the same. If, however, adding mass causes the object to become more compressed (smaller radius), then it could conceivably cause the object to speed up.

 

Yes, just adding mass (lets say the radius changes insignificantly) will slow the planet's rotation. Like you said you require that angular momentum be conserved.

 

You are also right in saying that compressing the planet would speed up the rotation. Again conservation of angular momentum demands this.

[csmyth3025]

I noticed that the quantity I arrived at in my post #3 calculation (2512 kg*m^2/s) is equivalent to 2512 J times 1 second:

 

 

where N is the newton, m is the metre, kg is the kilogram, s is the second, Pa is the pascal, and W is the watt

 

I'm not sure how to interpret this. If 1 Joule is equal to 1 Watt*second, what is 2512 J times one second equal to? Did I make a mistake in my calculation?

 

I'm trying to figure out how much work it would take to bring the sphere I described to a halt using more familiar units, such as comparing it with the amount of energy consumed by a 100 Watt light bulb in one hour (0.1 kWh).

 

I may be confusing angular momentum with rotational kinetic energy:

 

If a rigid body is rotating about any line through the center of mass then it has rotational kinetic energy () which is simply the sum of the kinetic energies of its moving parts, and is thus given by:

 

 

where:

 

• ω is the body's angular velocity
• r is the distance of any mass dm from that line
• is the body's moment of inertia, equal to

(ref. http://en.wikipedia....Rotating_bodies )

 

If I use the same value I used previously for the moment if inertia (400 kg*m^2) and ω^2 instead of just ω, I would get:

 

= {(400 kg*m^2)(6.28/s)^2}/2 = 7,887.7 kg*m^2/s^2 = ~7,888 W·s

 

If I understand this formula correctly, the amount of energy it would take to slow the sphere I described down to a halt would power a 100-watt light bulb for about 1 minute and 19 seconds (if it could all be converted with 100% efficiency to electricity).

 

Does this sound right?

 

I'm still a bit confused about how angular momentum is applied to problems of this type. What sort of problems use angular momentum?

 

Chris

[ajb]

I'm still a bit confused about how angular momentum is applied to problems of this type. What sort of problems use angular momentum?

 

When ever we have a rotational symmetry quite generally angular momentum is conserved. In more mechanical problems whenever we have torque a "turning force" then angular momentum is not conserved. Torque is the rate of change of angular momentum, just like force is the rate of change of linear momentum.

[baric]

Would adding more mass to the planet to increase the gravity have any effect on the core? Would the increased gravity speed the rotation of the core and create a magnetic field?

 

If the mass you were adding was a significant quantity of Iron and impacted the surface with enough velocity so as to vaporize the crust to allow the newly-added iron to liquefy and sink to the core. At that point, you would have the necessary ingredients for an Earth-like magnetic field.

 

Watching Mars melt would make an amazing light show on Earth, at least until some of the larger impact fragments arrived to wipe us all out.