How did astronomers "measure a large (>40 degrees) misalignment between the black hole spin and the orbital spin" of MAXI J1820+70?

Question

Gizmodo's Funky Black Hole Is Spinning Tilted says:

A team of astronomers recently observed an askew black hole just 10,000 light-years from Earth. The black hole’s spin axis is out of alignment with the plane of debris orbiting it by at least 40 degrees, the most extreme misalignment of its kind yet seen by astronomers.

The system is called MAXI J1820+70, and it contains a small black hole (about eight times more massive than the Sun) and a star that’s about half the size of the Sun. Astrophysicists recently calculated that the black hole, an incredibly dense object with a gravitational field so intense that not even light can escape, is spinning at an awkward angle compared to stuff orbiting it; the researchers think its tilt may be a relic of how the object was born, in the violent death of a star.

“The main result is that, for the first time, we measure a large (>40 degrees) misalignment between the black hole spin and the orbital spin,” said Juri Poutanen, an astrophysicist at the University of Turku in Finland and the paper’s lead author, in an email to Gizmodo. “In this case, the likely formation channel is a large kick (due to asymmetric neutrino emission) to the black hole during the collapse of the core of a massive star.”

and then links to the Science Perspective article A crooked spinning black hole which links to Juri Poutanen et al. (2022) Black hole spin–orbit misalignment in the x-ray binary MAXI J1820+070 which seems to have an arXiv preprint as Extreme black hole spin--orbit misalignment in X-ray binary MAXI J1820+070

Question: How did astronomers "measure a large (>40 degrees) misalignment between the black hole spin and the orbital spin" of MAXI J1820+70?

How direct/indirect is this "measurement" of >40 degrees? Is it something that can be visualized in an image somehow?

Answer 1

See Figure 3 from https://arxiv.org/pdf/2109.07511.pdf:

In this picture, the North and East directions are in the plane of the sky, and the $\hat{o}$ direction is in the direction of the observer. In blue, $\hat{\Omega}$ is the spin axis of the binary orbit. In red, $\hat{s}$ is the axis of the black hole spin. The $i$ angles are the angles between the line of sight $\hat{o}$ and the spin axes, and are referred to as inclinations. The $\theta$ angles are the angles between the North direction and the spin axes, and are referred to as position angles. The rest of this answer will follow the arXiv preprint and focus on the $i$ and $\theta$ angles as the 4 key angles that are needed to estimate the misalignment $\beta$.

Before this paper, other works had made measurements of $i_{jet}$ and $\theta_{jet}$. Simply put, the inclination of the jet affects how blobs of jet gas move across the sky (proper motion) and the position angle of the jet can be seen in images. Simulations show a "magneto-spin alignment" (Mckinney et al 2013) effect which align the jet direction with the black hole spin (shown in red above).

The orbital inclination $i_{orb}$ "has been constrained ... by the lack of x-ray eclipses and the detection of grazing optical eclipses." In simpler terms, a $i_{orb}$ too close to 90 degrees would produce x-ray eclipses since we'd be seeing the orbit close to edge-on. On the other hand, there were some observations where it appeared that the donor star just barely passed in front of the accretion disk (grazing optical eclipses) so the orbital inclination has to be large but not too close to 90, resulting in a range of possibilities between 66-81 degrees. You can see more details here.

The paper itself measures polarization angles and hypothesizes that these polarization angles might either be parallel or perpendicular (funnily enough) to $\theta_{orb}$. After trying many possibilities, most possibilities have misalignments greater than 40 degrees, but there is a lot of uncertainty.

Each of the 4 angles are derived by making (usually justifiable) assumptions but nonetheless a layperson would probably say these are indirect measurements that cannot be really "seen." The most obvious angle is $\theta_{jet}$ since you can see which direction the jet is pointed on the sky (just as you might see which direction Orion's bow points on the sky), but $\theta_{jet}$ is just a tool to estimate $\theta_{bh}$.

"Directness" is a very relative thing especially in astronomy. I think this measurement of misalignment is very interesting because each of the 4 angles required a different approach to estimate, and then they must all be statistically combined in the end to provide an estimate of misalignment.