math gets lit (with neutron stars)

This week's #mathematicalmodel is Emma, an astrophysicist, PhD student at the University of Southampton, and science communicator. If you remember from Phoebe's blog post, partial differential equations were useful in creating mathematical models of the ocean's food web. This week, they pop up again – only this time, they are used to model the behavior of neutron stars! Keep reading to learn about how neutron stars make gravitational waves by growing mountains, and how Emma uses mathematics to study them.


Neutron stars are incredibly compact stars that are formed in supernova explosions. They are the size of a city and about 1.4 times heavier than the sun. Neutron stars are the densest known stars made of matter in the universe. They have incredibly strong magnetic and gravitational fields, making them excellent laboratories for testing the laws of physics in extreme environments. The closest neutron star is between 250-1000 light years away, so how are we able to use them to test science if they are so far away? With maths of course! The great thing about maths and physics is that it doesn’t matter where you are in the universe, the rules are always the same.

Last week, the LIGO Scientific Collaboration announced gravitational waves had been detected from two neutron stars crashing into each other over 130 million light years away. This detection is incredibly exciting! For the first time ever both gravitational waves and electromagnetic waves were observed together. This led to many new discoveries. The centuries long mystery of where the heavier elements in the universe came from was solved as both gold and platinum was seen in the neutron star collision.

 Image by Oliver Dean at

Image by Oliver Dean at

Neutron stars don’t have to crash into each other to make gravitational waves, they can make them by changing shape and this is where my research comes in…

To make gravitational waves you just need some asymmetry, mass and acceleration. By this definition, that means humans should be able to produce gravitational waves and we do, so why don’t we try to detect these? The problem with gravity is that it’s weak, which means gravitational waves are even weaker. The gravitational waves we make are so small that it is physically impossible to ever detect them! To overcome this, scientists look for gravitational waves in parts of the universe where the strongest gravitational fields are seen, around black holes, neutron stars and supernova explosions.

Neutron stars are incredibly compact making them the roundest objects in the universe. To make gravitational waves they need to change shape, and they can do that through a process called accretion. Accretion occurs when a neutron star has a companion star similar to the sun. The neutron star’s gravitational field is so strong it pulls matter from the sun like star onto its surface. As the stardust falls onto the neutron star it releases gravitational potential energy in the form of X-rays. We can see these X-rays on earth using telescopes. About 5% of all neutron stars are thought to be accreting matter from a companion star.

 Image by Oliver Dean at

Image by Oliver Dean at

Accretion can be quite uneven, causing more stardust to fall onto the neutron star in some places than in others. Because this matter burns, hot spots can form in the neutron star crust. Hot spots are needed for mountains to grow. Nuclear reactions that usually occur deep inside a neutron star’s crust start to occur near the top of the crust when a hot spot is present. Mountains form when these nuclear reactions happen at the top of the crust. As the neutron star rotates, the mountains ‘hook’ onto spacetime, stirring it up, causing ripples to form just like ripples in a pond, except these stretch and squeeze space and time, and travel outwards at the speed of light.  These ripples are gravitational waves. Incredibly, for a neutron star to produce a gravitational wave signal that is detectable here on Earth, one of these mountains has to grow just a few millimetres high!

maths board.JPG

In my work, I solve the heat equation using a mixture of pen and paper maths (analytical) and computer simulations (numerical). The heat equation is a partial differential equation. It varies with both time and space. I setup up the heat equation analytically, by adding a heat source term (which comes from accretion), and a heat sink term (which comes from the heat being thermally conducted out of the star). The final equation is too complicated to solve by hand, so I code it up in Python and get the computer to calculate the answer. To check the computer has calculated the solution correctly I have to test my code. If my test results converge, I know it’s working, and if they diverge something is going wrong! The aim of my work is to calculate how hot these hot spots can be, how long they last for, and how big the mountains can grow.

I love the way Emma can take complex concepts of outer space, and explain them in ways that anyone can understand! She does more of this on her YouTube channel, "The Extraordinary Universe," so make sure you check out her vids!  Also follow her on Instagram at @emmanigma_, and check out her site at for more on her work!