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How do astronomers find out the correct weight of a planet tough there isn't any direct means to weigh them?What technique do they use?Just curious!

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Try to guess! Here a hint: Gravity... (not the movie) –  Py-ser Aug 13 at 4:55

1 Answer 1

First Weight and Mass are 2 different things.

In everyday usage, the mass of an object is often referred to as its weight though these are in fact different concepts and quantities. Mass refers to the amount of "matter"(or in lay word stuff) in an object, whereas weight refers to the force experienced by an object due to gravity.

We can calculate the Mass of the Stars, Galaxies or Orbiting Sun not Weight, because each object in space applies a gravitational force on every other body. So defining Weight of a body like a star is difficult (it will be the resultant gravitational force acting on it)


To calculate the mass of binary stars, stars with planets orbiting around it, we use Kepler's Third Law to calculate the mass of the star.

Kepler's third law states that, "The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit."

If the size of the orbit (a) is expressed in astronomical units (1 AU equals the average distance between the Earth and the Sun) and the period (P) is measured in years, then Kepler's Third Law says:

Kepler's Third Law

the trick to finding mass of star lies within the proportionality constant.

After applying Newton's Laws of Motion and Newton's Law of Gravity we find that Kepler's Third Law takes a more general form:

Kepler's Third Law

where, M1 and M2 are the masses of the two orbiting objects. and P is measured in seconds (all values are should be in same system like SI or CGS).

Note that here by substituting the values for Orbital Period, Pi, Gravitational Constant and length of semi major axis; we can calculate the quantity (M1 + M2) i.e. total mass of the system. If the mass of one body, such as M1, is much larger than the other, then M1+M2 is nearly equal to M1.

Hence to calculate mass of one particular star, the mass of planet/2nd binary star (in case of binary system) orbiting it should be small compared to the mass of star we want to calculate. But in case of planet orbiting a star, this condition is satisfied and we can calculate the mass of star easily as the mass of planet is very small compared the mass of star. e.g. in case of Sun-Jupiter System, the mass of Sun(1.988435×10^30Kg) is 1000 greater than Jupiter(1.8987 10^27Kg).

So, Measuring the mass of stars in binary systems is easy. Binary systems are sets of two or more stars in orbit about each other. By measuring the size of the orbit, the stars' orbital speeds, and their orbital periods, we can determine exactly what the masses of the stars are. We can take that knowledge and then apply it to similar stars not in multiple systems.

Example:

Phobos orbits Mars with an average distance of about 9380 km from the center of the planet and a rotational period of about 7hr 39 min. Use this information to estimate the mass of Mars.

Solution:

a = 9380 km = 9.38 x 10^6 m

P = 7 hr 39 min = 7.65 hr = 27540 sec

Calculation

Since the mass of Mars is so much greater than the mass of Phobos, (M1 + M2) is very nearly equal to the mass of Mars, so this is a good estimate.


Second Method: By measuring Luminosity of Star.

We can measure luminocity and temperature of any star. A plot of luminocity versus temperature for a set of stars is called a Hertsprung-Russel (H-R) diagram [https://en.wikipedia.org/wiki/Hertzsprung%E2%80%93Russell_diagram], and it turns out that most stars lie along a thin band in this diagram known as the "Main Sequence".

H-R Diagram

[This file is licensed under the Creative Commons Attribution-Share Alike 2.5 Generic license. Author: Richard Powell]

HR diagram showing many well known stars in the Milky Way galaxy

[This photograph was produced by European Southern Observatory (ESO). Their website states: "All ESO still and motion pictures are released under the Creative Commons Attribution 3.0 Unported, unless the credit byline indicates otherwise." Author: European Southern Observatory]

In H-R diagram, stars arrange themselves by mass on the Main Sequence, with massive stars being hotter and brighter than their small-mass bretheren. If a star falls on the Main Sequence, we therefore immediately know its mass.

Also by using the Mass-Luminosity equation we can calculate the mass of the star. The Mass-Luminosity equation is, Mass-Luminosity Equation

Where, where L⊙ and M⊙ are the luminosity and mass of the Sun and 1 < a < 6. The value a = 3.5 is commonly used for main-sequence stars. This equation and the usual value of a = 3.5 only applies to main-sequence stars with masses 2M⊙ < M < 20M⊙ and does not apply to red giants or white dwarfs.

Now by substituting the value of luminosity of star (which can also be found with help of H-R diagram), the luminosity and mass of the Sun and value of a, we get mass of that star.

The value of 'a' in Mass-Luminosity equation is determined using stars position in HR Diagram (or considering the range of mass of stars in which it belongs).

Also, our models of stellar structure are excellent predictors of the properties and evolution of stars. As it turns out, the mass of a star determines its life history from day 1, for all times thereafter, not only when the star is on the Main Sequence. So actually, the position of a star on the H-R diagram is a good indicator of its mass, regardless of whether it's on the Main Sequence or not.

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