Our journey begins here…

Bathed in the sunset’s glow, Arthur Dent enjoyed his pasta. His eyes were caught by a “Black Hole Chef’s Pasta House” ad in the Star Evening paper. This restaurant is said to have the universe’s supreme dining experience, where the chef uses black hole tidal forces to create unmatched hand-pulled spaghetti, with ingredients ranging from celestial bodies to stray astronauts, showcasing dishes that reflect the chef’s masterful control over matter and gravity. Let’s follow this whimsical journey with Arthur to the heart of culinary and cosmic wonder, where we will find ways to understand the mysterious and beautiful phenomenon of spaghettification near black holes.

A SPECIAL EVENT

Tidal force refers to the gravitational difference exerted by one celestial body on different parts of another body. A rough real-life model of concept is ocean tides. According to the formula for the tidal force, we know that its magnitude decreases with the cube of the distance between objects. Since Earth is a large sphere, the gravitational force exerted by the Moon is stronger on points of the Earth closer to the Moon than points farther away, which will cause Earth to deform and stretch towards the Moon. Since this gravitational stretching is negligible compared to Earth’s vast size, it is difficult to directly observe in Earth’s solid crust and mantle. Instead, the evidence is more clear in the oceans. Ocean water continuously bulges towards and away from the Moon, creating the ocean tides we observe.¹As shown in Figure 1, when the Moon and Earth line up horizontally, the ocean forms a bulge at each end of the Earth. 

In the universe, all bodies exert tidal forces on each other, influenced by their mass and proximity. Even though the Moon is near, its lightweight only produces small tides on Earth. In contrast, as one of the few heavyweight objects in the universe, black holes can exert extreme gravitational forces, leading to a much more dramatic effect called spaghettification. As the name suggests, the massive gravitational force of a black hole is not as simple as causing ocean tides but rather has a destructive effect on objects that get too close. 

To understand this process, we first need to know two characteristics of black holes: the event horizon, and the singularity. The event horizon refers to the gravitational boundary of a black hole, a vacuum solution of Einstein’s field equations developed by the German astronomer Karl Schwarzschild, suggesting that once inside this interface, nothing, not even light, can escape. The singularity represents the center of the black hole, where the entire mass of the star is compressed into a narrow space, making it a point with infinite gravity. With this information, we can imagine that the gravity on an astronaut’s feet heading towards a black hole would be far greater than that on their head, allowing the black hole to stretch them.² As the astronaut approaches the event horizon, the gravitational difference they feel also increases. As they cross the event horizon and approach the singularity of the black hole, the gravitational gradient becomes exceptionally steep, stretching the object infinitely into a thin, elongated shape. The laws of physics as we know them break down at the singularity point.

BIG OR SUPER BIG

While it is a common idea that all black holes can induce spaghettification, in more detail  this effect also depends on the type of black hole involved. Astronomers classify black holes into three main types based on their mass scale: stellar-mass black holes, intermediate-mass black holes, and supermassive black holes. Among them, stellar types are the smallest, with masses about 20 times that of the Sun. They are widely distributed, perhaps as many as a hundred million in the Milky Way alone. On the other hand, supermassive black holes are the heaviest, which are about hundreds of thousands to billions of times the mass of the Sun, existing at the centers of nearly every large galaxies.³ Figure 2 presents a comparison chart detailing the relative masses of the universe’s most dense objects.

Recalling Newton’s universal gravitation equation again, supermassive black holes undoubtedly have the largest gravitational field due to their incredible mass. However, it’s worth noting that spaghettification is not just about “huge gravity” but more about “huge gravitational gradient.” Therefore, it is necessary to return to the topic of the event horizon since it is the dividing line of the gravitational boundary of a black hole. Karl Schwarzschild’s calculations revealed that for a static, spherically symmetric celestial body. If all the mass is compressed into a sphere, it would create a surface where the escape velocity equals the speed of light known as the event horizon. This sphere’s radius is defined as the Schwarzschild radius.⁴ According to the formula rs=2GMc2, the Schwarzschild radius is directly proportional to the object’s mass. Hence, a more massive black hole has a larger Schwarzschild radius, which also implies that its event horizon is further from the singularity.

To show more precisely how tidal forces work near black holes with a focus, it’s essential to know that the tidal force experienced by an object when it’s near a black hole is inversely proportional to the cube of the distance from the black hole’s center (R) and directly proportional to the mass of the black hole (M). Therefore, the formula for tidal force can be expressed as FtidalMR3.  Additionally, the radius of the event horizon (the Schwarzschild radius, Rs) is linearly proportional to the mass of the black hole (RsM), follows that the tidal force observed at the event horizon scales inversely with the square of the mass of the black hole. This inverse relationship implies that doubling the mass of the black hole results in a fourfold decrease in the tidal force at the event horizon. This conclusion highlights the significantly weaker tidal forces exerted by supermassive black holes compared to their stellar-mass counterparts.

This framework allows further analysis of the variance in tidal forces across black holes of differing scales. Stellar-mass black holes are less massive, therefore possess relatively small Schwarzschild radii. Objects nearing these black holes experience intense gravitational differentials at the event horizon, causing them to stretch into elongated forms. Conversely, supermassive black holes, with their vast Schwarzschild radii, allow objects more time crossing the event horizon, subjecting them to a milder gravitational gradient.

A simple mathematical model provides insight into this distinction. Consider a meter-long, 80 kilogram object nearing both a stellar-mass and a supermassive black hole. To use an analogy, these forces are compared to the gravitational pull felt by a weight suspended from one’s feet. For a stellar-mass black hole (around 10 solar masses), the Schwarzschild radius is approximately 30 kilometers. At this event horizon, the tidal force on this object is equivalent to hanging a weight of roughly 800 million kilograms from one’s feet. Conversely, for a supermassive black hole (around one million solar masses), the Schwarzschild radius expands to about 3,000 kilometers. At this event horizon, the tidal force experienced by the same meter-long object would be akin to hanging a mere 80 grams (or the weight of a small apple) from one’s feet. These calculations confirmed the theory mentioned earlier: at the event horizon, tidal forces exerted by supermassive black holes are much weaker than those exerted by stellar-mass black holes.

However, this doesn’t mean that it is safe to approach a supermassive black hole. Approaching or crossing near the singularity, all matter inevitably undergoes intense spaghettification and becomes completely disintegrated.

 A STAR PASTA

So far, we have imagined and analyzed the phenomenon of spaghettification that a small, compact object would experience near a black hole. Now, let’s broaden our perspective to consider a larger-scale situation involving stars. Similar to how the Moon’s gravitational pull affects Earth, a supermassive black hole may exert a gravitational differential between the two sides of a nearby star. When this immense tidal force gradually overtakes the star’s own cohesive forces, the star is ultimately torn apart.

This is not mere speculation; in the 1990s, astronomers discovered these astronomical events when they observed a surge in electromagnetic radiation from the centers of galaxies, termed Tidal Disruption Events (TDEs). TDEs provide macroscopic confirmation of the spaghettification theory. During a TDE, a star is stretched and then destroyed, and the resulting debris is rapidly pulled towards the black hole, forming a spaghetti-shaped stream of material. The high-energy debris is ejected back into space, while the remaining low-energy material under the gravitational force of the black hole rotates towards it, forming a flat, disk-shaped structure called an accretion disk. The materials which reach the inner edge of the accretion disk will inevitably be swallowed by the black hole.^{5 6}

In October 2019, researchers at the University of California, Berkeley identified the closest observed instance of a TDE event. A star, similar in mass to the Sun, was spaghettified and torn apart by a supermassive black hole in a spiral galaxy within the Andromeda constellation. The brightness of the event allowed scientists for the first time to study this behavior in visible light. Interestingly, in this event, after the star being torn apart, most of left material did not fall into the black hole as expected but was ejected away from the black hole, leaving behind a new mystery to unravel.⁷

This interstellar journey to discover the delicious spaghetti has reached the end. We explored the theory of tidal forces, marveled at black holes’ boundless potential, and saw how they sculpt stars as if kneading dough. Looking ahead, the universe’s “kitchen” holds more secrets and surprises, waiting to be understood. New study will focus on observing these phenomena more directly, using the advancing technology of space telescopes and  gravitational wave detectors to get a closer look to the event horizons of black holes. The journey is far from over, and future discoveries promise even greater thrills in this extraordinary cosmic feast.

Acknowledgment

I would like to express my gratitude to Dr. Daniel Kasen, an associate professor of Physics, Astronomy Department and Dr. Eugene Chiang, a professor of Astronomy, Earth and Planetary Science Department. Their expertise and suggestions are significantly insightful and constructive, allowing me to enhance my presentation.

References

1. Newton, I. (1729). The Mathematical Principles of Natural Philosophy. Benjamin Motte.
2. Hawking, S. (1998). A brief history of time (Updated and expanded tenth anniversary ed). New York: Bantam Books.
3. Types of Black Holes. (n.d.). Retrieved April 1, 2024, from https://science.nasa.gov/universe/black-holes/types/
4. Gubser, S. S., & Pretorius, F. (2017). THE SCHWARZSCHILD BLACK HOLE. In The Little Book of Black Holes (pp. 44–74). Princeton University Press. https://doi.org/10.2307/j.ctvc774j3.6
5. Tidal Disruption – NASA. (n.d.). Retrieved April 1, 2024, from https://www.nasa.gov/image-article/tidal-disruption/
6. Komossa, S. (2015). Tidal disruption of stars by supermassive black holes: Status of observations. Journal of High Energy Astrophysics, 7, 148–157. https://doi.org/10.1016/j.jheap.2015.04.006
7. published, R. L. (2022, July 20). Cosmic crime scene reveals how black holes turn stars into “spaghetti.” Retrieved April 1, 2024, from Space.com website: https://www.space.com/black-hole-star-death-gas-cloud-clues

Image References

1. Observatory, E. S. (2020). Artist’s impression of star being tidally disrupted by a supermassive black hole [Photo]. Retrieved from https://www.flickr.com/photos/esoastronomy/50614978206/
2. NOAA Tides & Currents. (n.d.). Retrieved April 11, 2024, from https://tidesandcurrents.noaa.gov/restles3.html
3. Mass Chart for Dead Stars and Black Holes. (n.d.). Retrieved April 8, 2024, from NuSTAR website: https://nustar.caltech.edu/image/nustar141008b
4. de, S. (2018). English:  A simple illustration of a non-spinning black hole. Own work. Retrieved from https://commons.wikimedia.org/wiki/File:Black_hole_details.svg
5. information@eso.org. (n.d.). Hubble Finds Hungry Black Hole Twisting Captured Star Into Donut Shape. Retrieved April 9, 2024, from www.spacetelescope.org website: https://www.spacetelescope.org/images/opo2301a/
6. Matsopoulos, Noirl. T. (2023). English:  Vera C. Rubin Observatory Panorama. Vera C. Rubin Observatory Panorama. Retrieved from https://commons.wikimedia.org/wiki/File:Vera_C_Rubin_Observatory_Panorama_%28LSST-360-pano2-CC%29.jpg