Created by Davide Borchia
Last updated: Aug 24, 2022

Learn how to calculate the Schwarzschild radius, the point at which nothing can escape the gravitational pull of a black hole. Keep reading our article to discover one of the most famous, misunderstood, and interesting concepts of astrophysics. Here you will find out:

• What is the Schwarzschild radius;
• How to calculate the Schwarzschild radius;
• How to calculate the gravitational field of a black hole;
• Real results of the Schwarzschild radius formula;

And much more!

## What are black holes? Schwarzschild radius and other amenities

Black holes are mysterious astronomical objects that, thanks to their characteristics, entered the collective imagination: countless science fiction stories and movies revolve around black holes (though the number of stars revolving around black holes is way higher).

However, at least on the surface, black holes are not that hard to understand. You may have heard that mass deforms the space-time fabricwith the beautiful analogy of a trampoline and a bowling ball. A black hole is a region of space where we can find so much mass that the space-time is stretched so deep that nothing can escape it, not even light!

Black holes have a defined mass, squeezed into an extremely small spherical volume. If the radius of the sphere is smaller than a particular value, we witness two phenomena:

• The object becomes a "mathematical" black hole;
• The radius mentioned above becomes the distance at which nothing can escape the gravitational field of the black hole.

Every object has a Schwarzschild radius: if compressed to a dimension smaller than the one defined by its value, we would create a black hole! We say that the Schwarzschild radius is a property of every massive object: the equation for the black hole radius is always valid. However, for small objects, the required density gravitational force would clash with the fundamental nature of subatomic particles. Even though we can calculate the Schwarzschild radius for a black hole with the mass of a human, such an object couldn't exist (at least within our current understanding).

## What is the Schwarzschild radius: how to calculate the radius of a black hole

Mathematically speaking, the Schwarzschild radius is part of a solution to Einstein's field equations, for which we find a singularity.

The singularity corresponds to the event horizon: the equation for Schwarzschild radius allows us to find the value of the distance from the center of a non-rotating black hole from which light can't escape. The concept of the event horizon is not far from the one we define on Earth: learn more with our Earth curvature calculator

🙋 The event horizon is so-called since we can't observe anything happening behind it. Event horizons are complex theoretical objects, and astrophysicists still have much to learn about them. For now, know that you can cross them; doing so is not safe for you, and that you should watch the homonymous movie.

The equation for the Schwarzschild radius is:

$r_{\text{S}} = \frac{2\! \cdot \! G \!\cdot\! M}{c^2}$

Where:

• $r_{\text{S}}$ is the calculated Schwarzschild radius;
• $G$ is the gravitational constant;
• $M$ is the mass of the object; and
• $c$ is the speed of light.

🙋 Astrophysics has a problem with measurement units: dealing with massive and fast objects led them to use weird units such as solar masses and light years: that's what you can use in our Schwarzschild radius calculator too!

## How to calculate the gravitational field of a black hole

The gravitational field of a black hole at the Schwarzschild radius is an interesting quantity: it defines the acceleration due to gravity calculated where not even light can escape anymore. We use the equation for the gravitational field, with the value calculated at the Schwarzschild radius of the black hole:

$g = \frac{G\cdot M}{r_{\text{S}^2}}$

As you can imagine, this quantity is humongous: let's check some examples.

## The formula for Schwarzschild radius in action: calculating Schwarzschild radius in various astronomical objects

The Sun is an average star, even though special for us humans. Its mass is:

$M_\odot = 1,\!989\times 10^{30}\ \text{kg}$

Yup, it's a lot! But we will see way worse. 😉

What's the Schwarzschild radius of the Sun? Calculate it with our Schwarzschild radius calculator, or apply the equation for the event horizon:

\begin{align*} r_{\text{S}} &= \frac{2\! \cdot \! G \!\cdot\! M}{c^2} = \\ & = \frac{1}{\left(299,\!792,\!458\ \frac{\text{m}}{\text{s}}\right)^2}\\ &\left(2\! \cdot\!6.67408\!\times\!10^{-11}\ \frac{\text{m}^3}{\text{kg}\cdot\text{s}^2}\right.\\ &\left.1,\!989\times 10^{30}\ \text{kg}\right)=2,\!954\ \text{m} \end{align*}

The event horizon of a black hole with a mass equal to the one of the Sun would not even stretch $3\ \text{km}$. Earth's Schwarzschild radius is slightly more than $8\ \text{mm}$. Such black holes are too small to exist... as far as we know!

At the center of our galaxy lies a supermassive black hole. Sagittarius A* — this is its name — mass is a whopping $4.154\times10^6$ times the mass of the Sun. Using the Schwarzschild radius formula, we obtain a radius of the event horizon of:

$r_{\text{S}} = 12.271\cdot10^{6}\text{ km}$

Sagittarius A* fits comfortably in the orbit of Mercury. The object featured in the famous first picture of a black hole, the supermassive black hole at the center of the galaxy M87, has an event horizon radius of $128.35\text{ AU}$: four times the distance from Sun to Pluto. Needless to say, we are glad our stellar neighborhood is rather unassuming.

Davide Borchia
Mass
Suns
mi
Gravitational field
m/s²
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