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Using Gravity to Determine Mass Lecture Notes A101H February 12, 2009 Surface Gravity Surface gravity: the acceleration due to gravity at a planet’s surface F = GMm R2 = ma Since m appears on both sides of the equation we can divide by m to obtain acceleration surface gravity = GM = g R2 Surface Gravity for other Planets Surface gravity g = GM R2 Jupiter has a mass of 318 Earth masses. But its much larger radius means g is only 2.36 times that of Earth. (Jupiter has no solid surface; I used its radius as the level in the atmosphere where the pressure is 1 Earth atmosphere) Mercury has a dense iron core, so it has a higher surface gravity than Saturn’s moon Titan, although both are almost the same size Since M = (4/3)πρ R3 , where ρ is the mean density, then one can write g = (4/3) π Gρ R Surface Gravity Relative to Earth’s • • • • • • • Jupiter Neptune Earth Saturn, Uranus, Venus Mars, Mercury Titan Moon 2.36 1.14 1.00 0.91 0.38 0.14 0.17 How far does Earth’s gravity extend? We can use this equation to compute the acceleration due to Earth’s gravity at larger distances. Remember that R is always the distance to the Earth’s center g = 9.8 m/s2 at Earth’s surface The Space Station orbits at 300 km; R = 6378 + 300 km g = 8.9 m/s2 at the Space Station The Moon’s distance is approximately 60 Earth radii g = 0.27 cm/s2 at the Moon’s orbit => Remember, the acceleration is the same for any mass, m Newton’s form of Kepler’s 3rd law (M + m) P2 = 4π2 a3 G Often M >> m Mass of the Sun >> mass of a planet (Msun) P2 = 4π2 a3 G Newton’s form of Kepler’s 3rd Law • Newton’s form of Kepler’s 3rd law can be applied to any 2 bodies in orbit around each other. If M and m are nearly equal, one obtains the sum of the masses. If M >> m, as is the case for planets orbiting the Sun or for a moon orbiting a planet, one can neglect m and determine the mass of the larger body if P and a are known. Thus, Newton’s form of Kepler’s 3rd law is an important means of determining the mass of a distant body. • => Remember, only for a planet orbiting the sun, one can use simply P2 = a3 , with P in Earth years and a in AU Using velocity to determine mass One can also use the velocity of an orbiting object to determine mass. This is particularly useful when observing the motions of stars in our galaxy or other galaxies, where the orbital period would be millions of years. (Msun) P2 = 4π2 a3 G We know P = circumference of orbit / average velocity For a low mass object orbiting in a circular orbit about a large mass P = 2π x Radius / velocity, Substitute this for P in the above equation and one has the result V2 = G M R for the case of uniform circular motion. You could use this to find the velocity of the Space Station in orbit around the Earth Mean Orbital Speeds of the Planets We call this pattern “Keplerian rotation”, where there is a large central mass and smaller masses in orbit The orbital speeds of stars and gas in our galaxy are very different from Keplerian rotation, meaning that there is lots of hidden mass in the outer part of the galaxy, despite the massive black hole at the center Modern View of Gravity • In the context of general relativity, our modern view of gravity is that the presence of mass distorts the curvature of spacetime, altering the path that objects follow through 4-dimensional spacetime. • The fact that Earth orbits the Sun means that spacetime is curved in the vicinity of the Sun • A black hole is a “hole” in spacetime, a mass so concentrated that not even light can escape from it