SPACE TECH

How to measure planetary masses

How to measure planetary masses

The characteristics of the planets of the solar system have been known since the Middle Ages, at the time of Kepler and Galileo. That is, the masses of the planets could be approximately determined even with simple methods and instruments. In modern astronomy there are several methods to calculate the characteristics of planets, stars, clusters and galaxies.

An interesting fact: 99.9% of the total mass of the solar system is concentrated in the Sun itself. All planets together account for no more than 0.01%. Of this 0.01%, in turn, 99% of the mass falls on the gas giants (90% of which is on Jupiter and Saturn alone).

Calculate the masses of the Earth and Moon :
To measure the masses of the planets in the solar system, it is easiest to first find values for the Earth. As we recall, the acceleration of free fall is determined by the formula F=mg, where m is the mass of the body and F is the force acting on it.

Comparing these two formulas, and knowing the value of the gravitational constant 6.67430(15)-10-11 m³/(kg-c²), we can calculate the mass of the Earth. We know the acceleration of free fall on the Earth, 9.8 m/s2, and also the radius of the planet. Substituting all the output data, we get about 5.97 x 10²⁴ kg.

Knowing the mass of the Earth, we can easily calculate the parameters of the other objects in the solar system – the Moon, the planets, the Sun, and so on. With the Moon in general, everything is quite simple. Here, it is sufficient to take into account that the distances from the centers of bodies to the center of masses are inversely related to their masses. Substituting these figures for the Earth and its satellite, we get the mass of the Moon as 7.36 × 10²² kilograms.

Now we turn to methods of measuring the masses of the planets in the Earth group – Mercury, Venus, and Mars. After that we will consider the gas giants, and finally – exoplanets, stars and galaxies.

General techniques for determining planetary masses :
The most classic way to find the masses of planets is to calculate using the formulas of Kepler’s third law. It states that the squares of the periods of revolution of the planets are related in the same way as the cubes of the semi-major axes of the orbits. Newton refined this law a bit, adding to the formula the masses of the celestial planets

In this way we can find the masses of all the planets in the solar system and the Sun itself. Both the periods of revolution and the semi-major axes of the orbits of the planets of the solar system can be easily measured by astronomical procedures, accessible even without complicated instruments. And since we have already calculated the mass of the Earth, we can substitute all the digits into the formula and find the final result.

As for exoplanets and other stars (but only binary ones), astronomy usually applies the method of analysis of visible disturbances and oscillations. It is based on the fact that all massive bodies “perturb” the orbits of others.

Planets Neptune and Pluto were discovered by such calculations, even before they were detected visually, as they say “at the tip of the pen”.

Masses of planets in the solar system :
So, we have understood the general techniques for calculating the masses of the different celestial bodies and have calculated the values for the Moon, Earth and the Galaxy. Now let’s rank the planets in our system according to their mass.

Topping the list with the largest mass of the planets in our solar system is Jupiter, which was one order away from making our system double. A little more and we could have had two suns, the second one instead of Jupiter. Thus, the mass of this gas giant is 1.9 × 10²⁷ kg.

Interestingly, Jupiter is the only planet in our system whose center of mass rotation with the Sun is outside the surface of the star. They are separated by about 7% of the distance from the Sun’s surface.

The second most massive planet is Saturn, its mass is 5.7 × 10²⁶ kg. Neptune is next – 1 × 10²⁶. The fourth most massive planet, the gas giant Uranus, whose mass is 8.7 × 10²⁵ kg.

Then there are the planets of the Earth group, rocky bodies, in contrast to the gas giants with their large radius and relatively low density.

The heaviest of this group is our planet, its mass has already been calculated. Next is Venus, the mass of the planet is 4.9 × 10 ²²⁴ kg. After Mars in the evaluation, it is almost 10 times easier – 6.4 × 10²³kg. And it closes it, as a planet of the smallest mass, Mercury – 3.3 × 10²³kg. Interestingly, Mercury is still lighter than the two satellites of the solar system, Ganymede and Callisto.

Determination of masses of stars and galaxies :
To find the characteristics of individual star systems, the gravimetric method is used. Its essence is to measure the gravitational redshift of the star’s light. It is measured according to the formula ∆V=0.635 M/R, where M and R are the mass and radius of the star, respectively.

We can also indirectly calculate the mass of the star from its visible spectrum and luminosity. First, its luminosity class is determined using the Hertzsprung-Russell diagram, and then the mass/luminosity relation is calculated. This method is not suitable for white dwarfs and neutron stars.

The mass of galaxies is calculated primarily from the rotational velocity of its stars (or simply the relative velocity of the stars if it is not a spiral galaxy). Newton’s universal law of gravitation tells us that the centrifugal force of the stars in a galaxy can be expressed in a formula:

Only this time we substitute in the formula the distance from the Sun to the center of our galaxy and its mass. In this way we can calculate the mass of the Milky Way, which is 2.2 × 10⁴⁴g.

Don’t forget that this figure is the mass of the galaxy without taking into account stars whose orbits lie outside the Sun’s rotational orbit. Therefore, for more accurate calculations, we take the outermost stars in the arms of spiral galaxies.

For elliptical galaxies, the method for finding mass is similar, only there we take the relationship between the angular size, the velocity of the stars, and the total mass.

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