Lecture 8: Mass and Moment of Inertia

Mass and Moment of Inertia The masses of the planets were not known until the 17th century, when Galileo, Kepler and Newton laid the groundwork for understanding the dynamics of planetary bodies. In 1686, using Kepler’s laws of planetary orbits and his own laws of gravity, force and acceleration, Newton calculated that the mass of Jupiter was approximately 312 times larger than that of Earth (G was not known so it was not possible to calculate an absolute value for either mass). Modern observations show that this estimate is correct to better than ~3%. The dimensions of the planets are more difficult to determine than the mass. They were first estimated from the occultation of stars by the planets and later by spacecraft data, radar-ranging and Doppler tracking from spacecraft. From the planetary mass and radius, the average densities of the planets are easily determined, but the distribution of mass, as a function of radius, is almost totally unconstrained. The first estimates for the radial mass distribution within the planets were obtained from the moment of inertia of the planet rotating about its own axis. Typically, computing the moment of inertia depends on the fact that the planets are not spherical, but bulge outward at the equator due to the accelerations associated with rotation. In addition, the rotation axis of a planet is generally not perpendicular to the plane in which it rotates around the sun (or around a planet if it is a moon). This means that there is a torque on the non-sperhically symmetric part of the rotating planet and the axis of rotation precesses like a gyroscope (only the motions are small). As for a gyroscope, the rate of precession of the rotation axis depends on the moment of inertia of the planetary body, so that the moment of inertia can be estimated from this precession rate. Moments of inertia can also be calculated from the quadrapole moment of the gravity field of a planet, but such data are only available for Earth, Moon, Mars and Venus. The moment of inertia, together with the planetary mass, is useful for computing how mass is concentrated toward the center of a planet, and in particular in estimating the core size and/or density. The moment of inertia for a sphere of uniform density is I = (2/5)MR2, where M is the mass of the sphere and R is the radius. If the moment of inertia is smaller than (2/5)MR2, the planet is denser near the center than near the surface. We can also write the moment of inertia in terms of density (ρ) rather than mass: I=(8/15)πρR5. If instead of a uniform density sphere, we have a sphere with a uniform density spherical “core” with density ρχ and radius rχ, and a uniform density mantle of density ρm, then the moment of inertia and the mass are: I= 8" 8" #m R 5 + # $ % # m ) r$ 5 ( 15 15 M= 4" 4" #m R 3 + # $ % # m ) r$ 3 ( 3 3 It is typical for the moment of inertia of a planet to be expressed as a dimensionless number L=I/MR2: ! 5 5 # 8" &, ) m R + ( ) * + ) m ) r* / I 1 L= = % (. MR 2 $ 15 '.MR 2 10 This simplifies to L=2/5 for a uniform sphere (for rχ=0 or rχ=R). If the main density ! variations within the planet are due to a denser core and a less dense mantle, then L will be less than 0.4. And, if either the core density or the mantle density can be estimated, then the approximate radius of the planetary core can be determined from the mass and moment of inertia of the planet. It can sometimes be useful to write L as: 5 5 " 2 % ( m R + ( ( ) * ( m ) r) I L= =$ ' MR 2 # 5R 2 & ( m R 3 + ( ( ) * ( m ) r) 3 or ! ! 5. + " 2 %-1+ ( ( ) / ( m *1)( r) /R) 0 I L= =$ ' MR 2 # 5 &-1+ ( ( ) / ( m *1)( r) /R) 3 0 , / If we know the mass of the planet, then the minimum value of L will occur for a value of rx that can be determined by setting the derivative of L, with respect to rx, equal to zero. Earth Venus Mars Mercury Moon radius (km) average density (kg/m3) L 6378 6052 3396 2440 1738 5515 5243 3933 5427 3350 .3308 .33 .376 .33 .394 You can find these and other fun planetary facts at: http://nssdc.gsfc.nasa.gov/planetary/factsheet/