Download Using Gravity to Determine Mass

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