How Is The Mass Of A Planet Calculated?

You can't put a planet on a scale, so astronomers weigh one by watching how it tugs on something else, usually one of its moons or a passing spacecraft. Once you know the orbit's radius (r) and period (T), Newton's form of Kepler's Third Law gives you the planet's mass directly: M = 4π²r³ / (GT²), where G is the gravitational constant. For planets without a moon (Mercury, Venus), we instead measure how much they deflect a passing probe.

Planets are massive celestial objects, so physically weighing them is practically impossible. It’s not like we have a humongous planet-sized weighing scale just lying around! Therefore, the only way to do this is through a theoretical approach. Sounds a bit tough… how many of you, the knowledge-seekers, would volunteer to undertake such a cosmic mission?

The theoretical approach to determining the weight of the planet involves the laws of physics, and as it turns out, the approach isn’t really that complicated. The secret to the calculation, as you may expect, mostly lies in mathematics.

The Science Of Weighing

When you stand on a scale, the scale measures how strongly the Earth’s gravity is pulling you. Now, when it comes to measuring the weight of the planet, scale not only relies on the gravitational pull that the planet exerts, but also the mass of the planet itself. The heavier the planet, the greater its gravitational pull. So, scientists can weigh planets by measuring how hard they pull on other celestial objects. Let’s see how the planet’s weight is calculated using different approaches.

Before we begin, there needs to be one clarification. What astrophysicists often calculate is the ‘mass’ of the planet and not the ‘weight’. Yes, many of us use these terms interchangeably, but scientifically they are quite different. The mass measures how much matter is present in an object under consideration. On the other hand, weight measures how heavy the object is in a given gravitational context. To better understand this, you need to think about astronauts on a moon. Out there, he would feel much lighter, but his mass would remain the same. It’s simply that the pull of gravity on the moon is much less than what is experienced on Earth. So yes, mass is what they usually endeavor to measure.

Using Newton’s Law

To evaluate the mass of a planet, such as Earth, we can use Newton’s Law of Universal Gravitation. Using this, we know that the force of attraction between two objects is proportional to the product of their masses divided by the square of the distance between their centers of mass. In order to simplify the calculation part, we assume that their geographical centers are their centers of mass (the point where the body’s center of gravity is located).

Since we know the radius of Earth, we can use Newton’s Law of Universal Gravitation to calculate the mass of Earth in terms of the gravitational force it has on an object, i.e., its weight on the Earth’s surface, using the radius of Earth as our distance variable.

Using Kepler’s Law

The history is actually the other way around. Johannes Kepler (1571–1630) preceded Isaac Newton by decades. Working from Tycho Brahe's painstaking naked-eye observations, Kepler published his three laws of planetary motion between 1609 and 1619, the third of which links a planet's orbital period (T) and its average distance from the Sun (r): T² ∝ r³. Nearly seventy years later, Newton showed in the Principia (1687) that Kepler's third law is a direct consequence of his own law of universal gravitation, and added the gravitational constant and the mass of the central body to give us the modern form: T² = (4π²/GM) r³. This is the equation that lets us turn an orbit into a mass. If you know the distance to a moon (r) and how long it takes to circle its planet (T), you can rearrange to get the planet's mass M directly.

To calculate the mass of a particular planet using Kepler’s third law, we first need to know how far the planet is from Earth. This is generally done by bouncing signals off that planet and calculating the time it takes for the radar to return. Doppler radio is used for this purpose.

Thus, to calculate the mass of the planet, we need the distance of that planet from Earth, represented by R. Next, we need to know the orbital period of the moon, denoted by T. Finally, we need to know the largest angular separation of the planet and the moon, denoted by θ.

Kepler’s third law is given by the formula:

The radius of the earth, r, given in the formula above, can be calculated using the formula: r=Rθ

Rearranging the equation with Kepler’s equation:

If the moon is very small compared to the planet, we assume the moon’s mass to be nil and substitute a value of m=0 and directly get the mass of the planet. This method works for most of the moons in the solar system.

However, if the moon is relatively large, such as Charon, then we need to calculate the center of mass of the planet. The distance between the center of the planet to the center of mass of the planet and moon, denoted by d, can be subsequently used to calculate the mass of the planet using the equation: Md=m(r-d)

This gives the mass of the planet as:

Using Geometry

Another way to guess the mass of the planet is by applying principles of geometry. However, this involves more calculative guessing than the actual measurement. If you can calculate the volume of the planet and make a rough estimate about the density of the planet based on its composition, you can get a rough value of its mass. Remember, mathematically we know that the volume of the planet (sphere) can be calculated using the formula: Volume=(4/3)*π*(radius)³

By rearranging the density formula, we can calculate the mass of the planet from this equation: mass= density*volume.

What Is The Formula To Find The Mass Of A Planet (Worked Example)?

Let's pin down the actual numbers, because this is the part most people are really after. The cleanest way to "weigh" the Earth uses the gravity you already feel under your feet. Newton's law of universal gravitation says the pull on a small object of mass m sitting on the surface is F = G·M·m / R², where M is the planet's mass, R is its radius, and G is the gravitational constant. But that same pull is just the object's weight, F = m·g, where g is the surface gravity. Set the two equal, and the object's own mass m cancels out completely. Rearranging for the planet's mass leaves a beautifully simple result:

M = g·R² / G

Now plug in the measured values for Earth: g = 9.80 m/s², R = 6.37 × 10⁶ m, and G = 6.67 × 10^-11 N·m²/kg². That gives M = (9.80 × (6.37 × 10⁶)²) / (6.67 × 10^-11) ≈ 5.95 × 10²⁴ kg, comfortingly close to the accepted value of 5.97 × 10²⁴ kg. Notice that you need to know G first. That number stayed missing until 1798, when Henry Cavendish measured it with a delicate torsion balance, which is why his experiment is often described as the moment we finally "weighed the Earth". The very same M = g·R²/G works for any world whose radius and surface gravity you can measure, while the orbital method (M = 4π²r³/GT²) is what you reach for when you can watch a moon or probe circle it instead.

How Do You Calculate A Planet's Gravity And Your Weight On It?

The formula above runs just as happily in reverse. If you already know a planet's mass and radius, the surface gravity falls straight out of the same balance of forces: g = G·M / R². Drop in Earth's numbers and you get the familiar 9.8 m/s²; do the same for Mars (M = 6.42 × 10²³ kg, R = 3,390 km) and you get about 3.7 m/s², roughly 38% of Earth's. Jupiter, despite being over 300 times Earth's mass, only manages about 24.8 m/s² at its cloud tops, because its enormous radius (about 11 times Earth's) sits in the denominator and tames the result. Surface gravity, in other words, depends on mass and size, not mass alone.

Once you have g for a world, your weight there is simply your mass times that gravity (W = m·g). Your mass never changes, but the force the scale reads does. A person of 70 kg weighs about 686 newtons on Earth; on Mars they would weigh roughly 260 N (about 38% as much), and on the Moon, where g is only about 1.62 m/s², barely 113 N, around one-sixth of their Earth weight. That is exactly why your weight can vary from place to place even before you leave the planet, and why Apollo astronauts could bound across the lunar surface in their bulky suits.

How Do We Weigh Planets Around Other Stars?

Planets orbiting distant stars have no moon we can track from here and no spacecraft flying past, yet astronomers still weigh them, by watching the star instead of the planet. A planet and its star both orbit their shared centre of mass, so a hidden planet makes its star trace a small circle, wobbling toward us and then away. As the star approaches, its light is shifted slightly toward the blue end of the spectrum; as it recedes, the light is shifted toward the red. This is the radial velocity method, and the size of the wobble reveals the planet's mass, since a heavier planet tugs its star harder.

A companion technique, the transit method, watches a planet cross in front of its star and dims the starlight by a tiny, measurable amount, which reveals the planet's radius. Put the two together and you get both mass (from the wobble) and size (from the transit), and dividing one by the other gives the planet's density, the clue that tells astronomers whether a world is a rocky body like Earth or a puffy ball of gas. It is the same logic ScienceABC describes for studying exoplanets we can never visit: read the starlight carefully, and the planet gives up its secrets. Whether it is the Earth beneath us or a world light-years away, weighing a planet always comes back to one idea, measuring the gravity it commands.

Who knew that the seemingly herculean task of measuring a planet's weight could be done so cleanly by applying the basic laws of physics and mathematics! Today, the most precise planetary masses come from spacecraft: NASA's Juno mission has refined Jupiter's mass and gravity field since arriving in 2016, and the BepiColombo mission (in Mercury orbit by late 2026) will sharpen our value for Mercury's mass too. Even exoplanet masses, light-years away, are measured the same way, by tracking how each world tugs its star or its sibling planets.