Research Interests
My research interests lie in computational astrophysics, with my current focus on non-ideal MHD processes in star formation. Some of my interesting results are highlighted below:
Low-mass star formation: Isolated
Low-mass star formation: In low-mass clusters
Low-mass star formation: Initial conditions from low-mass clusters
Disc formation and fragmentation
Extreme ionisation rates during the formation of the Larson core
NICIL: Non-ideal mhd Ionisation Coefficients & Ionisation Library
SPH: Smoothed particle hydrodynamics

Click here to view all my research videos on my YouTube page.

Low-mass star formation: Overview
Stars are the lifeblood of the Universe. They provide light so that we can study kinematics of galaxies. They process gas to enrich the interstellar medium. They host planets upon which life may live. And when massive stars die, they collapse into black holes. Therefore, understanding stars is incredibly important for understanding all aspects of astronomy. Since a star's evolution is dictated by its mass at birth, understanding the star formation process and, by extension, star forming regions, is therefore also incredibly important for all aspects of astronomy.

The general star formation process has been understood for nearly a half a century. First, a cloud of gas initially undergoes an isothermal collapse due to the low optical depths and the long wavelength of the radiation. As the central region becomes more dense, it begins to efficiently trap radiation, and evolves almost adiabatically producing a pressure-supported object known as the first hydrostatic core; this object has a typical radius of 5au and a few Jupiter masses. The first core continues to accrete material from its surrounding envelope until its central temperature reaches ~2000K; at this point, molecular hydrogen begins to dissociate, triggering a second collapse phase. Once the hydrogen becomes mostly atomic, a second hydrostatic core, also known as the stellar core forms; the core continues to accrete material from the remaining envelope to produce a young star.

The phases of star formation.

During this process, the conservation of angular momentum results in the formation of a protostellar disc, known as a Class 0 disc. When exactly this disc forms is uncertain since they are observationally challenging to observe at this phase in their life, and simulations suggest they can form as early as in the first core phase or well into the stellar core evolution phase. Outflows are also launched during the star formation process, with slow outflows typically being launched during the first core stage and fast outflows during the stellar core evolution phase.

Magnetic fields permeate star forming regions, and they act to support against gravitational collapse. Therefore, if the magnetic field strength is too strong, then the gas cannot collapse and a star cannot form! It is expected that gravity is only 2-3 times stronger than the magnetic field, suggesting that the gas will collapse, but the magnetic field will play a role in star formation.

The simplest description of magnetic fields is ideal magnetohydrodynamics (MHD), where it is assumed that the gas is mostly ionised and the ions are tied to the magnetic field; under this assumption, there is no resistivity. Realistic star forming regions are only weakly ionised, thus physical resistivity exists between charged and neutral species. The three important non-ideal MHD processes for star formation are
Ohmic resistivity: the drift between electrons and ions/neutrals; neither ions nor electrons are tied to the magnetic field,
Ambipolar diffusion: ion-neutral drift; both ions and electrons are tied to the magnetic field, and
Hall effect: ion-electron drift; only electrons are tied to the magnetic field.

A simple graphical summary of ideal MHD and the various non-ideal MHD terms.

Different effects or combination of effects are important during the different phases of star formation. Ohmic resistivity and ambipolar diffusion are both dissipative, and locally weaken the magnetic field and allow neutral gas to slip through the field. Compared to ideal MHD, this results in less pinched magnetic field lines and weaker magnetic field strengths in the central regions of collapsing cores, yet similar gas densities; see the following figure. The dissipative processes act on small scales, thus the magnetic fiels lines are qualitatively similar on the large scales.

Magnetic field lines superimposed on density (red frame) and magnetic field strength (blue frame) slices during the first hydrostatic core phase for ideal and non-ideal MHD simulations.

The Hall effect is dispersive and hence dependent on the initial geometry. Specifically, the Hall effect produces a drift velocity that is perpendicular to the direction of the magnetic field. If the rotational and magnetic field vectors are parallel and aligned, then this drift will be in the opposite direction to the initial rotation and hinder disc formation. If the rotational and magnetic field vectors are parallel and anti-aligned, then this drift will be in the same direction as the initial rotation and promote disc formation. See the following figure.

The contribution of the Hall effect to the spin of an object.

Low-mass star formation
Contrary to what is expected from observations, star formation simulations that include ideal magnetic fields launch well-defined first- and stellar-core outflows, however, they do not form discs. Simulations that include non-ideal MHD and are initialised with their magnetic field vector anti-aligned with the rotation vector form large protostellar discs, but launch weak first core outflows and no stellar core outflows.

Below is a video of low-mass star formation using ideal magnetic fields, non-ideal magnetic fields with aligned magnetic field and rotation vectors, and non-ideal magnetic fields with anti-aligned vectors.

A collapsing, one solar mass gas cloud, modelled in the presence of a strong magnetic field. The left panel is modelled using ideal MHD, the centre panel models non-ideal MHD where the magnetic field and rotation vectors are aligned, and the right panel models non-ideal MHD where the magnetic field and rotation vectors are anti-aligned. The non-ideal MHD models use the canonical cosmic ray ionisation rate. The results are published in Wurster, Bate & Price (2018c).

When we model each non-ideal MHD term individually, then we find that the Hall effect has the greatest influence on the evolution of the star and its surrounding environment, as shown in the following video. More details can be found in Wurster, Bate & Bonnell (2021) and Wurster, Price & Bate (2016).

A collapsing, one solar mass gas cloud, modelled in the presence of a strong magnetic field. The left-hand video uses ideal MHD, the centre two columns include only one non-ideal process, and the right-hand column contains all three non-ideal processes. Given the Hall effect's dependence on the orientation of the magnetic field, both aligned and anti-aligned configurations are modelled. The non-ideal MHD models use the canonical cosmic ray ionisation rate. The left-most and right-most models are the same models as in the preceding video. The results are published in Wurster, Bate & Bonnell (2021).

By increasing the ionisation rate by even a factor of ten higher than the canonical rate yields results with no discs and weak outflows, as can be seen in the following videos, and discussed Wurster, Bate & Price (2018a).


A collapsing, one solar mass gas cloud, modelled in the presence of a strong magnetic field. The top video shows the evolution of the gas density in a slice through the code, and the bottom video shows the evolution of the magnetic field in a slice through the core; the left-hand model uses ideal MHD and the right-hand model uses non-ideal MHD with a cosmic ray ionisation rate ten times higher than the canonical value. The results are published in Wurster, Bate & Price (2018a).

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Low-mass star formation in low-mass clusters
Stars are born in chaotic environments, unlike the idealised environments used above. As described above, however, these regions are permeated with magnetic fields and are weakly ionised, thus are best modelled by non-ideal MHD. The movie below shows the effect of the magnetic field strength on the formation and early evolution of low-mass clusters. The large scale structure is clearly affected by the field strength, with more filamentary structures in the strong-field cases and clumpy structures in the weak field cases.
The formation and early evolution of low-mass star clusters containing 50 solar masses of gas. All models include non-ideal MHD and are initialised with vertical magnetic fields. The initial magnetic field strength decreases from left to right; the structure changes from filamentary to clumpy from left to right.

From our studies on the formation of isolated stars, the evolution is clearly dependent on the inclusion/exclusion of non-ideal MHD. On these low-mass cluster scales, however, the structure is approximately independent of the non-ideal MHD processes. On small scales differences do appear, with tighter filaments and weaker magnetic fields in the non-ideal models.
The formation and early evolution of low-mass star clusters containing 50 solar masses of gas. The models are initialised with vertical magnetic fields of moderate strength and include non-ideal MHD (left) and ideal MHD (right). The large scale structure is similar between both models, while the small scale structure in the clumps and around the stars varies.

Finally, protoplanetary discs form in all of our models, suggesting that there is no magnetic braking catastrophe. Thus, the catastrophe explored during the formation of isolated stars is likely a result of idealised initial conditions, and disappears once more realistic environments are modelled. The following image shows each star in each of our models, clearly showing that discs form even in clusters initialised with strong, ideal magnetic fields.

Gas column density around every star in our study; all the discs are orientated to be face-on, and the target star is placed at the centre of the frame. Each white dot represents the location of the star. Not all stars have discs, but a variety of circumstellar, circumbinary and circumsystem discs form. This suggests that there is no magnetic braking catastrophe. This is Fig. B1 of Wurster, Bate & Price (2019).

For more details, please see Wurster, Bate & Price (2019) or view the remaining videos.

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Low-mass star formation: Initial conditions from low-mass clusters
What is the nature of a star forming clump? Observations reveal these to be chaotic environments being modified and influenced by many physical processes. However, numerical simulations often define these initial star forming clumps to be idealised objects. To better understand numerical star forming environments, and to create realistic initial conditions for future studies, in Wurster & Rowan (2023) and Wurster & Rowan (2024), we analysed and evolved star forming clumps extracted from the low-mass clusters of Wurster, Bate & Price (2019).

In Wurster & Rowan (2023), we extracted and analysed 109 clumps, which are shown below:


An animation of all the clumps we extracted from Wurster, Bate & Price (2019). The animation cycles through the clumps from all nine clouds, showing both the extracted clumps themselves and the parent cloud for direct comparison. For static images, click here.

We found that, within each clump, the gas density distribution was not smooth and spanned 2-4 orders of magnitude. The clump magnetic field was ordered, but not reflective of the initial magnetic field geometry of the parent cloud. In general, most clump properties had a slight trend with clump mass but were independent of (or only very weakly dependent on) the properties of the parent cloud. We concluded that stars are born from a wide variety of environments and there is not a single universal star forming clump.

In Wurster & Rowan (2024), we evolved 29 of the extracted clumps from two different cloud simulations. The evolution of the 17 clumps extracted from a model that evolved non-ideal magnetic fields are shown below:

The evolution of the 17 clumps extracted from the non-ideal MHD low-mass star cluster simualtion (Wurster, Bate & Price (2019)) with an initial normalised mass-to-flux ratio of 5 formation. The panels appear at the time the clump was extracted from the parent cluster. For a similar video for the clumps extracted from an ideal MHD simulation, click here. Videos with different spatial scales and employing ideal MHD are available at this playlist.

To determine the reproducibility of the systems that spawned the clumps, we compared their evolution to the evolution of the counterpart gas from the parent cloud.

A comparison of the evolution of the 17 extracted clumps (top row in each group) compared to the evolution of the counterpart region of the star cluster itself (bottom row in each group). For a similar video for the clumps extracted from an ideal MHD simulation, click here. Videos with different spatial scales are available at this playlist.

We were unable to identically reproduce any of the systems in the cluster simulations, however, we qualitatively reproduced a few systems, and statistically reproduced the stellar and disc properties (except for the magnetic field strength). Despite failing to identically reproduce the systems in the parent cluster, we showed that our clumps provided excellent initial conditions that evolved to produce a range of stellar systems and disc morphologies. Therefore, using extracted clumps to model star formation rather than idealised initial conditions provides greater insight into the star formation process and the associated physical processes.

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Disc Formation and Fragmentation
In Wurster & Bate (2019), we performed 105 simulations to investigate disc formation and fragmentation. We investigated the effect of initial rotation rate, magnetic field strength, magnetic field orientation, and the inclusion of the non-ideal MHD processes. Most model form discs, although those with slow rotations and strong, ideal magnetic fields are susceptible to the magnetic braking catastrophe and do not form discs. Discs are more likely to fragment in the presence of fast rotations and weak magnetic fields. The image below shows the evolution of the gas column density of each model in our study.


The evolution of the gas column density of each model in Wurster & Bate (2019). Frames become black at the end of the simulation. The penultimate frame shows the final data we have for each simulation. The ultimate frame shows each simulation at its end time (either when the disc dissipates, fragments, or at approximately 16kyr after the formation of the disc).

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Extreme (cosmic ray) ionisation rates
Increasing the cosmic ray ionisation rate should make a non-ideal MHD model approach an ideal MHD model, while decreasing the ionisation rate should make a non-ideal MHD model approach a a purely hydrodynamical model. In Wurster, Bate & Price (2018b), we find that models with high ionisation rates can be equivalent to ideal MHD models; models with low ionisation rates are never equivalent to hydrodynamical models, but they do approach them.

The following images show a density cross-section for models with varying cosmic ray ionisation rates at three different maximum densities. The progression from ideal MHD through reasonable ionisation rates to purely hydrodynamical is clear. Click on the image for additional plots of magnetic field strength, rotational and radial velocities.

Density slices parallel to the axis of rotation through the core of the clouds at three different maximum densities. The cosmic ray ionisation rate decreases from left to right. Blank frames indicate missing data due to computational limitations. (Click image for additional plots of magnetic field and velocity slices.)

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NICIL: Non-Ideal mhd Coefficients & Ionisation Library
Nicil is a stand-alone Fortran90 module that calculates the ionisation values and the coefficients of the non-ideal magnetohydrodynamics terms of Ohmic resistivity, the Hall effect, and ambipolar diffusion. The module is fully parameterised such that the user can decide which processes to include and decide upon the values of the free parameters, making this a versatile and customisable code. The module includes both cosmic ray and thermal ionisation. Version 2.1 includes 6 cosmic ray reactions and 30 chemical reactions that involve neutral gas species (H, H2, He, C, O, O2, Mg, Si, S, CO, HCO), positive ions (H+, H3+, He+, C+, O+, O2+, Mg+, Si+, S+, HCO+), electrons, and positively charged, negatively charged and neutral dust. The default dust prescription is a single grain size, however, options for the MRN distribution or a user-defined distribution are included.

The source code to the module is publicly available from BitBucket.org and the reference papers are Wurster (2016) and Wurster (2021). Nicil has been thoroughly tested in SPMHD codes, but is written to be platform independent, thus can also be implemented into grid codes. The following figures show the non-ideal MHD coefficients and their constituent components plotted against density and temperature. Both sets of plots use a barotropic equation of state, thus are drawn from the same data set.

The species number densities (first column), the conductivities (second column) and the non-ideal MHD coefficients (third column) as calculated by Nicil version 2.1 using a barotropic equation of state.

The library is under continual development, so please consult modifications.pdf and IMPLEMENTATION.txt in the home directory of the repository for details of how the code has evolved from that described in Wurster (2016) and Appendix A of Wurster (2021).

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SPH: Smoothed particle hydrodynamics
My research involves using and developing the smoothed particle hydrodynamics (SPH) method, specifically, the (compressible) smoothed particle magnetohydrodynamics (SPMHD) flavour. I use both Phantom (of which I am a lead developer) and sphNG.

This method was first published in 1977 by Lucy (1977) and Gingold & Monaghan (1977). Since then, it has been greatly modified to incorporate new physics, and is now used not only in world-leading astrophysical studies, but also widely used in engineering and the entertainment industry.

The following may be useful resources:

My presentations include several videos and animations that do not display on these slides.

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