THIS UNIVERSE
PROF YASH PAL
How do scientists measure or calculate the weight of a planet? Firstly let me clarify that we do not measure the weight of a heavenly body but only its mass. Your weight changes when you move from the surface of the earth to that of the moon, because the force of gravity changes. Your weight in a space orbit might become zero! But the mass does not change. Your weight is the force of gravity on you and mass is the quantity of matter that determines the inertia you offer when subjected to a force. It is a deep riddle of nature that weight and inertial mass are proportional to each other. Indeed living on the surface of the earth we often forget the difference between the two. One is still trying to find whether there is a miniscule difference between the two. Let us now come to your question, reformulated as “how do we determine the mass of a planet”. One way is to find out the force of gravity on a test object placed at a given distance. This is not practical for distant planets. Instead we use the fact that if the planet has a satellite at a known distance from it then the period of revolution of that satellite is a unique function of the mass of the planet. The only approximation that we need to use is that the mass of the satellite should be very tiny compared to that of the planet. In the following, allow me to use a couple of simple equations though they are not necessary for a qualitative understanding. For the satellite to be a satellite the force of gravity towards the planet should be balanced by the centrifugal force of circular motion (for simplicity we use only circular orbits for this argument). This means that G M m/r2 = m v2 /r or v2 = G M /r or M = rv2 G The period of the orbit is T = 2nr/v. Here G is the known universal gravitational constant, and r can also be the known distance of the satellite. It is then easy to solve for M the mass of the planet by putting in the value of v in terms of the orbital period and r. Notice that the small mass of the satellite, “m”, does not enter into consideration in this simple treatment. This is the way we know the mass of the sun, the earth and the other planets. There is no other way. All we have assumed is that if there is mass there must exist a corresponding gravitational field. Since I have used a few simple equations you might think that I have given something complicated. Spend a little time on this and you will find
that it is not so. To summarise, the mass of the sun is determined by analysing the periods of rotation of various planets. The mass of the earth could have been estimated by observing the period of the moon but much more accurately now by the orbits of artificial satellites. Masses of some of the other planets were also estimated by analysing the motions of their satellites. For some of the planets where proper natural satellites are not available we use the perturbations of orbits of artificial space probes. The basis for all this is the belief that gravitational constant is universal, that mass must result in a gravitational field whose variation with distance is also known. It might be interesting to know that that increased sophistication of measurement and calculation techniques has lead to the discovery of black holes and a large number of extrasolar planets. But the basic physics remains the same. In principle high school students could have made these discoveries! What is optical density? Optical density is, I believe, the power of refraction of light. It is usually given in terms of a number called refractive index. Some materials are more optically dense than others, for example diamond is far denser than water. Some glasses called lead glasses are also denser than ordinary glass. The refraction arises from the fact that the velocity of light is reduced inside the medium in comparison to its velocity in vacuum. In fact refractive index of a material is the ratio of velocities in vacuum and that in the material. Refractive index of water is 1.33 while that of diamond is 2.42. Preciousness of diamond for jewelry comes from this property. In a finely cut diamond a lot light incident on its surface is total internally reflected from various surfaces because of the high refractive index.
