You can measure the mass of something that orbits a larger body by how far away they are, and how long they take to make one complete orbit. Johannes Kepler, in the 1500's, predicted that the orbital sizes and periods of the planets should have a very definite proportion - that is: If we call T = time to make one orbit around the sun, and R = radius of the orbit, then (R³)/(T²) for all the planets = a constant, which includes the mass of the sun. So, to measure the mass of a planet, you can measure the periods and radii of the orbits of all the moons that orbit around it , graph R³/T² for all the moons, and calculate the mass of the planet. This is one way to do it.

Another way is by measuring the shape and the gravitational field, and then you can calculate the mass.

This is a good question. What astronomers actually calculate is the mass of all the planets, not their weight. The mass tells you how much matter there is in the planet but the weight tells you how heavy something is on the Earth. The difference is that, on the moon, you are much lighter than on the Earth, but you still have the same mass (because you have the same amount of matter in you!).

Anyway, the way to calculate the masses of the planets is using Newton's law of gravity The law of gravity is an equation that tells you how strong objects pull on each other due to gravity, and it depends on the masses of the two objects and on their distance apart. Using this law of gravity, you can calculate exactly where the planet should be and how fast it should be going at any time, and all of that information turns out to depend on the mass of the sun and the mass of the planet. By studying the exact orbit of the planets and sun in the solar system, you can calculate all of the masses of the planets.

Yes. The masses of the planets are calculated most accurately from Newton's law of gravity, a = (G*M)/(r²), which can be used to calculate how much gravitational acceleration (a) a planet of mass M will produce on objects at distance r away.

One can solve for M once the other numbers are known. G is a universal number which determined here on Earth by measuring the attraction of objects of known mass at a known spacing r, then solving for G. (The law of gravity applies to all objects, no matter how small- we just don't notice the gravity of everyday objects, because it is so weak). The measurement is done with a Cavendish balance, which is surprisingly sensitive- I used one once, but got too close to the device and the experiment was ruined because of the gravitational pull of my head.

A planet's mass can in principle be calculated by observing how its gravity causes small wobbles in the orbits of other planets. In fact, college student John Couch Adams realized that small wobbles in Uranus's orbit could be nicely explained by the presence of an undiscovered planet. Astronomers looked where Adams told them to, and found Neptune (which other astronomers, including Galileo, had seen but dismissed as just another faint star). Since a planet's gravity is stronger near the planet, a more accurate determination of its mass involves observing the orbits of its moons. All planets except Mercury and Venus have moons, but those two planets have been visited by space probes, along with all other planets except Pluto. Probes provide the most accurate measurement of the mass because the probe's motion can be measured to high accuracy by recording the times at which its radio pulses arrive on Earth. The probe's distance from Earth determines how long it takes the signal to reach us, and with high-accuracy clocks we can measure, within meters, how much the probe was gravitationally deflected by the planet.

The short answer to your question is that yes, scientists calculate the mass of planets. Before describing it to you in detail, I should point out that in physics, the terms "mass" and "weight" have significantly different meanings. Mass is a measure of how much matter there is in an object. For any object, this number is the same anywhere in the universe. Weight, on the other hand, has to do with the attractive force between different objects. Every object with mass attracts every other body with mass according to Isaac Newton's equation
F = G * M * m / (r²) rwhere M is the mass of one object, m is the mass of a different object, d is the distance between them, and G is the "gravitational constant."

First off, we can see that as the mass of the objects increases, their product will increase and give bigger forces. However, G is a very, very small number (Side note: although G is extremely small, it is possible to measure it in a laboratory, as was originally done by Cavendish just before 1800). The other key part is that the attractive force (gravity) decreases as the inverse square of the distance between the objects. That means, if the distance between the two objects doubles, the force drops by 4; if it triples, the force drops by 9; if it's quadrupled, the force drops by 16, and so on.

So how do we use this to measure the mass of the planets? For this, there are basically two options - you either
1. Need to be on the planet, or
2. Get some information about an object orbiting the planet. Let's talk about each one.

1. Calculating the mass of a planet while on that planet.
It turns out that in the above equation, the group of terms
G*M / (r²)
is what scientists think of as the acceleration due to gravity (taking M as the mass of the planet). This is something that can be experimentally measured. For example, on Earth, the value is about 10 m / s². That means if I drop an object on earth (any object such that there is negligible air resistance), after 1 second, it will be falling at about 10 m / s; after 2 seconds it'll be about 20 m /s ; after 3 seconds it'll be about 30 m /s, and so on. So, if you drop an object on another planet, you could first measure how fast it accelerates towards the planet. Since you know what G is, and r would be the radius of the planet, you could then calculate M, or the mass of the planet!

2. Calculating the mass of a planet from earth
Since we can't fly to the different planets to use method 1, we are forced to come up with the mass of planets a different way. To do this, you need to have some information about an object that is orbiting it. Without going into details, it turns out that starting from the first equation above, you can describe an object (say, a moon) orbiting a another object (say, a planet) by the equation
M = 4 * pi² * r³ / T⁽ 2G)
Here, M is the mass of the planet, pi is 3.14..., r is the distance between the center of the moon and the center of the planet, T is the period (the time it takes the moon to make one full orbit around the planet), and G is again the gravitational constant. The interesting part about this equation is that you do not need to know the mass of the orbiting moon to get the mass of the planet! If we see an object orbiting a planet, then all we need to do is figure out how far apart they are, and how long one orbit takes, and we can calculate the mass of that planet!

The weight of the planet can be computed through several methods:

1) For our planet, the volume of the planet has been measured for several hundred years assuming it is a perfect sphere. Also the average density is known form measurements at the surface and in the depth of the earth. Once you have the 2 it is easy to compute the approximate weight of the earth.

2) For other planets it is more complex. Lately, radar waves are used to measure the density of the planet. The shape and dimensions of the planet are also measured using radar waves and astronomical observations. Once the shape, dimensions and density are known, it is trivial to get the weight.

3) Another technique is to use Newton's law which predicts the trajectory in space of a planet gravitating around another one. The trajectory can be measured precisely using astronomical techniques. The weight of one of the planet (eg. The earth) is known and using Newton's gravity law, and the known trajectory of the other planet (eg the moon) , its weight is deduced. I am sure there are many other techniques which I do not know.

Well, the best way to do it is to measure the orbit of something around the planet, like a satellite. If the satellite is small enough compared to that of the planet, the path it travels around the planet will depend on the mass of only the planet. Many of the planets have moons orbiting around them. All you have to do is watch the moon travel around the planet and figure out how far it is from the planet as it orbits and how long it takes to make one circuit around the planet. You also, of course, need to know the distance to the planet. That can be measured by observing the planet's orbit around the Sun and by using parallax methods (observing the planet "shifting" in the sky compared to the background stars when observing from different locations on Earth). In reality, it's a bit more complicated than I've said, but this is the basic idea.

Like much of astronomy, the masses of the planets are known by a set of linked assumptions about the universe. We know the mass of the earth since we can measure the gravitation constant (using a known mass). Since we also know the force of gravity on the surface of the earth, we can find its mass. Then, since we know the period of the revolution of the earth about the sun (1 year), we can find the mass of the sun as well. Finally, we know the periods of the planet orbits by observations from the earth -- so we can use the known solar mass to find the planet's masses. (The assumption is that newton's law of gravity applies everywhere...) If you think this is a long chain of reasoning, you should see how one finds the distance to far away galaxies and galaxy clusters...

You probably know that massive objects, such as planets, attract each other. This effect is called gravity. Newton first understood the mathematical equations that describe this attraction. Kepler later extended Newton's theories and derived an equation know as Kepler's law. This equation relates the distance between a planet and its moon to the period of the moon?s orbit, and the planet?s mass. Therefore, if you know the distance to the moon, and how long it takes the moon to travel around the earth, you can use Kepler's law to 'weigh' the earth.

The weight of planets is not easy to calculate, but we can get an idea with some simple geometry. If you know how big they are (i.e. treating them as a sphere, with a radius) then you can work out their volume (I think the rule is something like
Volume = 4*pi*radius²)
Once you have the volume, you have to make some assumptions about what sort of rock or gas the planet is composed of, and how dense this material is.

For instance, the radius of the Earth is about 6370 km. This is a volume of about 5,000,000,000 cubic kilometers. Most of the earth below 40 km deep is composed of an olivine rock called peridotite which has a density of about 3300 to 3500 kg per cubic meter. Therefore if we multiply this figure by 1000 to put it in cubic kilometers and multiply it by the volume of the Earth, we get a awesomely huge figure! This is only a low estimate though, as rocks become more dense as you go deeper into a planet (and the Earth is composed of different rock types with different densities, especially towards the core).

We don't know much about what sort of rocks are on Mars or Venus, hopefully this will become clearer over the next decade as we send more probe missions there.