Mathematical Model for Oddly Shaped Asteroids Useful to Spacecraft

By David DeFusco
Communications Manager
Katz School of Science and Health

Hektor, the largest Jupiter Trojan asteroid that has its own moon.

Three Katz School mathematics and physics researchers have developed a theoretical framework for predicting the possible shapes and gravitational fields of asteroids.

The results, published in the international journal Astrophysics and Space Science in March, can be useful for spacecraft engineers developing landing designs for irregularly shaped celestial objects. The research was funded by a $412,000 grant from the National Science Foundation.

“One of the paper’s central challenges was to find mathematical expressions for representing gravity on the surface of an irregularly shaped asteroid, understanding that, unlike on Earth, gravity on these objects isn’t constant,” said Dr. Fredy Zypman, a professor of physics and co-author of the paper “Surface Gravity of Rotating Dumbbell Shapes” with Dr. Marian Gidea, professor and chair of the M.S. and Ph.D. mathematics programs, and Dr. Wai-Ting Lam, a doctoral alum in mathematics and now a member of the faculty at Yeshiva University’s Stern College for Women.

Asteroids in the solar system can take on a variety of shapes. They are of particular interest to scientists and explorers because they’re rich in minerals. “The exploration of the irregular gravity fields is compelling,” said Dr. Lam. “In particular, dumbbell shapes are among those that have been observed for comets and asteroids.”

The Katz School researchers focused on dumbbell-shaped, or peanut-shaped, asteroids whose gravitational fields can vary widely on their surfaces because their mass is unevenly distributed, as opposed to Earth, a nearly rounded object that produces a relatively constant gravitational field.

“Dumbbells are among the shapes that have been observed for comets and asteroids, making them both astronomically and mathematically interesting,” said Dr. Gidea. “Because the gravitational field of an asteroid is complicated, more irregular, spacecrafts have to be very careful on their approach.”

Examples of oddly shaped asteroids include Hektor, the largest Jupiter Trojan asteroid that has its own moon; the Comet Hartley 2, which was the target of a flyby in 2010 by NASA’s Deep Impact spacecraft; and the trans-Neptunian Arrokoth, or Ultima Thule, located in the Kuiper Belt, which was the target of the New Horizons space probe’s flyby in 2019.

“In addition to the general results for gravity on peanut-shaped objects, we also created a model for the shape of Hektor that can be described by simple equations. This formula gives us a possible family of shapes for this type of asteroid,” said Dr. Gidea. “The peanut-shaped asteroids aren’t all identical. We had to figure out how many exist and the family of shapes, and then we figured out the gravity at any point in the vicinity of these shapes.”

Dr. Zypman said the researchers also studied how the shape of an asteroid changes depending on its rotation. “This knowledge is relevant for understanding how the asteroid formed initially, when the object was still malleable and approaching its current shape.”