2gud
November 18th, 2005, 04:58 AM
Going by echarcha's request Starting a separate thread for this interesting Physics info:
Given that you know the mass of earth M, distance between earth and moon R, time period of moons rotation T around earth, there is no way you can find the mass of the moon. This is because any body/rock of any mass which is at a distance R and revolving in the same (approx circular) trajectory of moon will have the the same time period T.
Because of the above reason and since M, R and T these are only the directly available values to a resident of earth, calculating moon's mass is a tricky problem.
The correct and by far most accurate way to calculate moon's mass is to put a satellite of any mass (contrary to my earlier claim, sorry for that but mass of satellite is not needed, I will shortly tell you why), in a near circular orbit of known distance (d) from moon 's center of mass and measure its period of revolution (t) around moon. Then we calculate moon's mass m from the relation:
t = 2*PI * Sqrt ( d ^ 3 / G * m)
As you can see time period of rotation of a satellite is directly proportional to square root of the distance cubed and inversely proportional to square root of the mass of parent planet
Also it is easy see that the above formula is identical to = 2*PI Sqrt ( L / g ) which is time period of oscillation of a simple pendulum, when we consider accln due to moon at satellite's location g = G*m / d^2. This is the beauty of Physical laws, the same law governing the motion a simple pendulum or a falling apple on earth governs moon also images/smilies/smile.gif
mass of moon then will be
m = ( 4*PI^2*d^3 ) / G*t^2
This is the most accurate way available to measure mass of any body in the universe, ony prerequisite is that we need to have at least a satellite orbiting it.
Just knowing two freely observable values - satellite's distance from the parent body and satellite's time period or revolution - we can determine mass of the parent body with the above formula.
One more case in point is astronomers were unable to determine the mass of the 10th planet (planet X or Xena) from the Sun till recently. Now they found a satellite Gabrielle orbiting planet Xena (yes they used characters from the legend of Xena!) and using the above formula they can now accurately determine Xena's mass. See here (http://www.universetoday.com/am/publish/10th_planet_has_moon.html?3102005)
Especially these lines -
"A combination of the distance of the moon from the planet and the speed it goes around the planet tells you very precisely what the mass of the planet is," explains Brown. "If the planet is very massive, the moon will go around very fast; if it is less massive, the moon will travel more slowly. It is the only way we could ever measure the mass of Xena-because it has a moon."
And ofcourse they won't know Gabrielle's mass unless it has its own satellite!
Can anyone suggest other means to calculate moon's mass? Even approximate methods would be welcome!
Given that you know the mass of earth M, distance between earth and moon R, time period of moons rotation T around earth, there is no way you can find the mass of the moon. This is because any body/rock of any mass which is at a distance R and revolving in the same (approx circular) trajectory of moon will have the the same time period T.
Because of the above reason and since M, R and T these are only the directly available values to a resident of earth, calculating moon's mass is a tricky problem.
The correct and by far most accurate way to calculate moon's mass is to put a satellite of any mass (contrary to my earlier claim, sorry for that but mass of satellite is not needed, I will shortly tell you why), in a near circular orbit of known distance (d) from moon 's center of mass and measure its period of revolution (t) around moon. Then we calculate moon's mass m from the relation:
t = 2*PI * Sqrt ( d ^ 3 / G * m)
As you can see time period of rotation of a satellite is directly proportional to square root of the distance cubed and inversely proportional to square root of the mass of parent planet
Also it is easy see that the above formula is identical to = 2*PI Sqrt ( L / g ) which is time period of oscillation of a simple pendulum, when we consider accln due to moon at satellite's location g = G*m / d^2. This is the beauty of Physical laws, the same law governing the motion a simple pendulum or a falling apple on earth governs moon also images/smilies/smile.gif
mass of moon then will be
m = ( 4*PI^2*d^3 ) / G*t^2
This is the most accurate way available to measure mass of any body in the universe, ony prerequisite is that we need to have at least a satellite orbiting it.
Just knowing two freely observable values - satellite's distance from the parent body and satellite's time period or revolution - we can determine mass of the parent body with the above formula.
One more case in point is astronomers were unable to determine the mass of the 10th planet (planet X or Xena) from the Sun till recently. Now they found a satellite Gabrielle orbiting planet Xena (yes they used characters from the legend of Xena!) and using the above formula they can now accurately determine Xena's mass. See here (http://www.universetoday.com/am/publish/10th_planet_has_moon.html?3102005)
Especially these lines -
"A combination of the distance of the moon from the planet and the speed it goes around the planet tells you very precisely what the mass of the planet is," explains Brown. "If the planet is very massive, the moon will go around very fast; if it is less massive, the moon will travel more slowly. It is the only way we could ever measure the mass of Xena-because it has a moon."
And ofcourse they won't know Gabrielle's mass unless it has its own satellite!
Can anyone suggest other means to calculate moon's mass? Even approximate methods would be welcome!