Astronomy 311: Tidal Forces• Assume the Earth and Moon are perfect spheres• Mass of Earth = 5.9376 × 1024kg, Moon’s mass = 7.3477 × 1022kg.• Radius of Earth = 6378.750 km, radius of Moon = 1737.4km.• Earth-moon distance (center to center) is 384403km.• The force due to gravity between two objects of mass M1and M2a distancer apart isF =GM1M2r2,where the unit of for ce is Newtons and G = 6.67 × 10−11m3kg−1s−2.• Calculate the gravitational force between the Moon and the Earth.• This a ssumes the Moon and Earth are both point particles. But they havea certain size.• Now calculate the gravitational force due to the Moon on a particle of mass1kg on the surface of the Earth, on a line joining the center of the Earthand Moon.• Now calculate the gravitational force due to the Moon on a rock of mass1kg lying 200km below the Earth’s surface a nd again on a line joining thecenter of the Earth and Moon.• Now calculate the gravitational force due to the Moon on a rock of mass1kg lying exactly at the center of the Earth.• Now calculate the gravitational force due to the Moon on a rock of mass1kg lying at the North Pole.• Now calculate the gravitational force due to the Moon on a rock of mass1kg lying at the South Pole.• Now calculate the gravitational force due to the Moon on a rock of mass1kg lying on the Earth’s surface as far away from the Moon as possible.• Relative to the center of the Earth, in what directions are the forces onrocks lying on the same and opposite sides of the Earth to the Moon.• So what do you think will happen to a perfectly shaped sphere orbiting theMoon?• Does the Earth produce a similar effect on the Moon?1• This effect, over a long period of time, leads to synchronous rotation, thatis the Moon orbits its own NS axis in a time equal to the time it takes forthe Moon to rotate around the Earth.• Why, then, doesnt the Earth spin about its NS axis in a time equal to thetime it takes to ro t ate around the Moon?• Tidal forces are due to gravity acting differentially on different parts of a2-body system.• In the Earth-Moon case, this leads t o tides (hence the name): two hightides in a 25-hour period as Earth rotates under an orbiting Moon.• Such tidal forces cause friction (tidal friction) which constantly decreasesthe Earth’s spin rate at the ra te of 0.0016s /century. This means the Moonis drifitng further away from the Earth at a rate of 3-4cm/year. Since theEarth’s rotat io n is decreasing, its angular momentum is decreasing. But theangular momentum of the whole Earth-Moon system must be conserved,so the distance of the Moon from the Earth must increase.• Extra-Credit: Can you use actual numbers to show this in a calculation?• Synchronous rotatio n is common throughout the Solar System: two Martianmoons, Galilean moons and most of Saturn’s moons.• Pluto and its moon, Charon, have reached the final stage of tidal evolution:both are in mutual synchronous rotation.• A detailed treatment of this topic suggests that tidal forces between twoobjects orbiting each other varies as 1/r3, where r is the distance betweenthem. So as a moon gets closer to its planet, tidal effects become moresevere: the shape of the moon becomes increasingly elongated.• The tidal acceleration on an a piece of 1kg rock on a Moon or biting a planetof mass MP, radius RP, is2GMPRm/r3,where r is the distance between the centers of the two worlds, MPis themass of the moon, RPis the radius of the planet, whereas the gravitationalacceleration of the same 1kg piece of rock to t he center of the Moon isGMm/R2m.• In some situations, the tidal force on the Moon due to its parent planet isgreater than the gravitational force holding the Moon together.2• For the Earth-Moon system, can yo u find a value of r, the distance betweenparent planet and moon that would make this so? In this situation theMoon would break up. That is find a value of r such that t he gravitationalforce on a 1kg of rock to the Earth is greater than the force to the centerof the Moon.• This gives the Roche tidal limit, or Roche limit or Roche zone. Saturn’sRoche limit is 1.24 × 108m. Most of Saturn’s rings are inside this limit.• Find the Roche Lobe limit for Saturn’s moon Titan. At what orbital dis-tance from Saturn is Titan