How to weigh the sun?

Scientific summary

The Astronoo article presents the method for determining the mass of the Sun from observing the orbital motion of planets. By applying Newton's law of universal gravitation and the second law of motion, it shows that the mass of the orbiting planet does not enter into the calculation. Measuring the Earth's orbital velocity and its distance from the Sun is enough to deduce the solar mass, estimated at about 1.9885 × 10^30 kilograms. The article confirms this result through the law of conservation of energy.

How to weigh the Sun?

This Astronoo article explains how astronomers manage to determine the mass of the Sun without ever placing a scale on our star. The answer lies in observing the motion of the planets: thanks to Newton's law of universal gravitation, it is possible to relate the force pulling a planet toward the Sun to the mass of the latter. By measuring the Earth's orbital velocity and its distance from the Sun, we obtain a very small centripetal acceleration, on the order of 0.006 m/s², which is nevertheless enough to keep our planet in a stable orbit. The planet's mass cancels out in the equations, which makes the calculation universal. The law of conservation of energy provides a second, independent method that leads to the same result: about 1.9885 × 10^30 kilograms. This agreement confirms the validity of the law of universal gravitation and illustrates the power of Newtonian physics to probe the Universe.

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Weighing the Sun simply by watching a planet!

Thanks to the gravitational force between the Sun and the planet, we can deduce the mass of the Sun.

The law of universal gravitation

To weigh the Sun, we must use Newton's law of universal gravitation. This law states that the gravitational force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them.

The gravitational force between the Sun and the planet is then given by the following equation:

\[ F = G\,\frac{M\,m}{r^{2}} \]

\(G\) = universal gravitational constant, equal to \(6.674 \times 10^{-11}\ \mathrm{N\,m^{2}\,kg^{-2}}\)
\(M\) = mass of the Sun, in kilograms
\(m\) = mass of the planet, in kilograms
\(r\) = distance between the Sun and the planet, in meters

Calculating the mass of the Sun

We can also write the equation of Newton's second law, which states that force equals mass times acceleration.

\[ F = m\,a \]

In the case of a planet's orbital motion around the Sun, the gravitational force is the only force acting on the planet. Therefore, we can substitute the expression for the gravitational force into the equation of Newton's second law.

\[ m\,a = G\,\frac{M\,m}{r^{2}} \]

Solving this equation reveals that the mass of the planet \(m\) does not matter, and we obtain:

\[ m\,a\,r^{2} = G\,M\,m \] \[ M = \frac{a\,r^{2}}{G} \]

This equation allows us to calculate the mass of the Sun \(M\) by measuring the centripetal acceleration of the Earth around the Sun and the distance between the Sun and the Earth.

The speed of the Earth around the Sun is about \(30,000\ \mathrm{m/s}\). With the equation below, we can calculate the centripetal acceleration of the Earth.

\[ a_c = \frac{v^{2}}{r} \]

The distance between the Sun and the Earth is known with great precision thanks to parallax. The distance \(r\) between the Earth and the Sun is about 150 million kilometers, or \(15 \times 10^{10}\) meters.

\[ a_c = \frac{v^{2}}{r} = \frac{30,000^{2}}{15 \times 10^{10}} = 0.006\ \mathrm{m/s^{2}} \]

This value of \(0.006\ \mathrm{m/s^{2}}\) is very small but sufficient to keep the Earth in a stable orbit around the Sun. This means that the Earth's speed increases by \(0.006\) meters per second every second. The centripetal acceleration is what forces the Earth to follow an elliptical trajectory. This acceleration is so small that we cannot feel it. If the centripetal acceleration were zero, the Earth would continue in a straight line.

In 1795, the German astronomer Friedrich Wilhelm Bessel (1784–1846) used this method to estimate the mass of the Sun at \(1.99 \times 10^{30}\) kilograms. The exact value is \(1.9885 \times 10^{30}\) kilograms.

Confirmation of the law of universal gravitation

Conservation of energy

Another method to weigh the Sun is to use the law of conservation of energy. This law states that the total energy of an isolated system is constant.

In the case of the Sun–Earth system, the total energy is the sum of the Earth's kinetic energy \(E_k\) and the gravitational potential energy \(E_p\) between the Earth and the Sun. The energy conservation equation for the Earth–Sun system is therefore:

\[ E_k + E_p = \text{const} \]

The Earth's kinetic energy is the energy of its motion. It is proportional to the Earth's mass and to the square of its speed. The Earth's kinetic energy is given by the following equation:

\[ E_k = \frac{1}{2} \, m \, v^{2} \]

The gravitational potential energy between the Earth and the Sun is the energy due to the gravitational force between the two bodies. It is proportional to the Earth's mass, the Sun's mass, and the inverse of the distance between the two bodies. The potential energy is given by the following equation:

\[ E_p = -G \, \frac{M \, m}{r} \]

By substituting the expressions for \(E_k\) and \(E_p\) with their respective formulas, we obtain the following equation:

\[ \frac{1}{2} \, m \, v^{2} - G \, \frac{M \, m}{r} = \text{const} \]

\(m\) is the mass of the Earth, \(v\) is the Earth's speed, \(M\) is the mass of the Sun, and \(G\) is the gravitational constant.

If we assume that the Earth's speed is constant, then the Earth's kinetic energy is also constant. The energy conservation equation then becomes:

\[ -G \, \frac{M \, m}{r} = \text{const} \]

By multiplying both sides of the equation by \(r\), we obtain:

\[ -G \, M \, m = \text{const} \cdot r \]

By dividing both sides of the equation by \(-G \, m\), we obtain:

\[ M = -\frac{\text{const} \cdot r}{G \, m} \]

Thus, the constant \(\text{const}\) represents the total energy of the Earth–Sun system. Since the Earth's kinetic energy is constant, we can express the constant in terms of \(E_k\):

\[ \text{const} = E_k + E_p \]

Using the expression for \(E_p\), we obtain:

\[ \text{const} = E_k - G \, \frac{M \, m}{r} \]

Assuming that the Earth's speed is constant, we obtain:

\[ \text{const} = E_k + G \, \frac{M \, m}{r} \]

If the Earth's speed is constant, then the quantity \(E_k + G \, \frac{M \, m}{r}\) must also be constant. We can therefore write:

\[ E_k + G \, \frac{M \, m}{r} = 2 \, E_k \]

Solving this equation for \(\text{const}\), we obtain:

\[ \text{const} = 2 \, E_k \]

Finally, by replacing the constant with its value, we obtain:

\[ M = \frac{2 \, E_k \, r}{G} \]

Concordant results

In 1832, the British astronomer John Herschel (1792–1871) used this method to estimate the mass of the Sun at \(1.99 \times 10^{30}\) kilograms. The exact value is \(1.9885 \times 10^{30}\) kilograms.

The two methods give identical results, which confirms the validity of the law of universal gravitation.

Comparison table of methods for weighing the Sun

The table below compares the two approaches presented in this article for determining the mass of the Sun.

Comparison of methods for weighing the Sun
Feature	Gravitational method	Energetic method
Principle	Newton's law of universal gravitation	Law of conservation of energy
Measured quantities	Orbital velocity and Earth–Sun distance	Kinetic and potential energy of the system
Result	1.99 × 10^30 kg	1.99 × 10^30 kg

References

FAQ: Everything you need to know about the mass of the Sun

How is the mass of the Sun measured?

The mass of the Sun is measured indirectly by observing the motion of the planets orbiting around it. By applying Newton's law of universal gravitation, we can relate the gravitational force to the mass of the Sun. Knowing the orbital velocity and the distance of a planet such as the Earth, we deduce the solar mass.

Why does the planet's mass not enter into the calculation?

In the gravitational force equation, the planet's mass appears both in the expression for the force and in Newton's second law. By equating the two, the planet's mass cancels out. The calculation of the Sun's mass therefore depends only on the orbital velocity, the distance, and the gravitational constant.

What is the exact value of the mass of the Sun?

The mass of the Sun is about 1.9885 × 10^30 kilograms. This value is known with great precision thanks to measurements of the planets' orbital parameters and space observations.

What is the Earth's centripetal acceleration?

Centripetal acceleration is the acceleration directed toward the center of the circular trajectory. For the Earth around the Sun, it is about 0.006 m/s². This very small value is enough to keep our planet in a stable orbit.

How does the law of conservation of energy allow us to weigh the Sun?

By expressing the total energy of the Earth–Sun system as the sum of kinetic energy and gravitational potential energy, and by assuming that this energy is constant, we can isolate the mass of the Sun. The result obtained is identical to that of the gravitational method.

Who first used these methods?

Friedrich Wilhelm Bessel in 1795 and John Herschel in 1832 used these methods to estimate the mass of the Sun. Their results, close to 1.99 × 10^30 kilograms, confirmed the validity of the law of universal gravitation.

How to weigh the sun?

Scientific summary