and . The symbols have their usual meaning, e.g. from Cunningham (1975), namely
There's a future blog post in my take on the physical significance of all of these symbols, but that's not for today.
It is a known result that circular orbits exist in the equatorial plane of static axis-symmetric spacetimes, i.e. where and , as shown by Carter (1968). We can further constrain our mathematical picture by noting that circular orbits have no motion in the radial components,
We can use the invariance of four velocities to express these constraints in terms of the metric,
where, for time-like geodesics, the effective mass parameter . In order to rearrange for the radial component, this equation may be expanded,
and, using the freedom to raise and lower indices under contraction, rewritten
In this form, the usual identification is permitted for energy and angular momentum from the covariant momenta,
The first constraint from eq. (4) may now be calculated from this expression, namely
If we rephrase the above constraint in terms of effective potentials, we find the expression cleans up to
Our constraints may be modified now to and , the latter of which is directly
calculated from eq. (9).
The term is omitted, as it is manifestly zero, since . Implicitly it is assumed that and are constant with respect to over an orbit. Indeed, this assumption is valid as and are constants of motion for any given geodesic, whereas the and which we later solve for are functions in the phase space. This distinction is subtle but important to note.
Here we will specialize to the Kerr metric, with subsequent manipulations performed using SageMath, and the occasional small bit of hand manipulation. Substituting in the metric components from eq. (2), we obtain
It is worth noting that has reappeared in the derivative term. This is due to the complexity of eq. (11) when taking the derivative in SageMath – instead, SageMath prefers to work with eq. (12) to produce more visually friendly results. Both will eventually solve the system identically, so the choice is largely stylistic.
In the equatorial plane, ; therefore
Noticing a similar term in the two equations, we can rearrange eq. (14) for
and substitute into eq. (15) to obtain
which may be simplified further:
Our constrains in eq. (4) are now eq. (14) and (18) respectively, which may be solved simultaneously to find and satisfying circular orbits. In SageMath, this is fiddly, and sometimes SageMath will fail after a minute, or run for seemingly hours without terminating. But one soon develops a sense for what SageMath can and cannot do, and after this, it is generally quite straight forward to solve, even for other metrics. We find there are four solutions in total, representing the intersection of the positive and negative energy regimes with prograde and retrograde orbits,
where the first denotes the positive and negative energies, and the second the prograde and retrograde respectively. The angular momentum can be rearranged as
At the time of writing, this is the most simplified form I have been able to manipulate these equations into, which seems pretty okay since we haven't had to even touch any advanced techniques.
As an additional note, with SageMath, many of the preceding steps in preparing the constraints are not actually necessary, but would provide results that are unwieldy for such a blog post. I was also partially curious to see if simplifying the constraint equations dramatically would make any difference to the solutions (it didn't really).
The Bardeen et al. paper quotes a more beautiful analytic solution
both of which are now ubiquitous in the literature.
They are also substantially different from eq. (19) and eq. (20). We can try to find the degree of similarity by plotting the equations with to avoid the central singularity:
The plots for energy (left panel) are in good agreement up until the energy becomes negative in the Bardeen case (though such energies are allowed in e.g. the Penrose process). The SageMath derivation fails here, presumably due to an ambiguity when squaring roots, which loses sign information. Indeed, we do have a negative energy solution from SageMath, but we would require a priori knowledge of which solution to apply in which domain. This wouldn't be ideal, and without a more refined solution, the SageMath approach would not even suggest negative energies are possible.
The consequence of the energy sign is also pronounced in the angular momentum (right panel). For the case, the plots are still in good agreement, but the divergence for is quite apparent, and worse still for (not depicted, as the diagram became difficult to annotate). Even when using the Bardeen et al. energy solution in the SageMath expression for , some of the features of the curves do not disappear, such as the sharp cusp at , which, again, is problematic.
Fortunately, however, the differences in the two solutions is for our purposes pedagogical, since the divergence lies within the photon orbit, at which radii stable circular massive orbits are long since prohibited.
There are many special boundaries in the Kerr metric. For purely circular orbits, we consider three of these boundaries and the respective possible orbits within them, as they are arguably most relevant to our study – these are the innermost stable circular orbits (ISCO), the marginally bound orbits, and the photon orbit.
There is of course also the event horizon, but this is not an orbital boundary. We will draw reference to this boundary, but only with the need of understanding that we cease to consider geodesics after they would pass this horizon.
The innermost stable circular orbit, as the name suggests, are the set of orbits which are both circular and energetically parabolic (stable). Stability requires the energy be a minimum; this may be rephrased as
which solves for an inequality , which is in the Bardeen paper. Physically, these are the orbits where perturbations to the potential will not cause the orbit to diverge; it is the energetically preferred state, if you will.
In direct contrast to the ISCO, we consider unstable circular orbits at . All such unbound circular orbits have hyperbolic energies (will diverge from circular orbits when perturbed), with a specific radius hosting the marginally bound orbits, defined as where . Any small perturbation of such a marginally bound orbit outwards will escape the black hole, but with asymptotically zero velocity.
The marginally bound radius is may be solved for the Kerr metric as
corresponding to first (infalling) radius at which the condition is satisfied.
Photons are null geodesics with , and, due to this, cannot form circular orbits anywhere except at a one particular radius, known as the photon radius, denoted . This radius is where , or, equivalently, where
i.e. the denominator from eq. (21) vanishes. This equation is solved by Bardeen, with the solution
The photon orbit represents, if you will, the innermost possible (unstable) circular orbit, which requires velocity equal to the speed of light. Orbits between with can only form unstable circular orbits, which may quickly and catastrophically collapse.
Analogous to the marginally bound orbits, the unstable orbits at the photon orbit, if perturbed outwards, will arrive at infinity still traveling at the speed of light.
With solutions for the energy and angular momentum, and a brief overview of relevant orbital boundaries, we can begin to simulate some of these orbits.
Tracing stable circular orbits when and is known is relatively simple. We only need to calculate , since our integrator automatically constrains via eq. (5). We find
using our eq. (19) and eq. (20) with for time-like geodesics. Setting up the integrator is then quite straight forward:
using GeodesicTracer using ComputedGeodesicEquations using GeodesicBase # define `energy` and `angmom` functions Δ(M, a, r) = r^2 + a^2 - 2 * M * r function energy(M, a, r) var = 2 * M * a * Δ(M, a, r) * sqrt(M * r) denom = r^5 - M * r^2 * (4 * Δ(M, a, r) + 2r^2 - M * r) en = ( 5 * M^2 * a^2 * r + 16 * M^2 * r^3 - 7 * M * r^4 + r^5 - 3 * (4 * M^3 + M * a^2) * r^2 - var ) sqrt(abs(en / denom)) end angmom(E, M, a, r) = E * a + (r / √M) * sqrt(abs(1 - r + r * E^2)) angmom(M, a, r) = angmom(energy(M, a, r), M, a, r) # function for calculating the initial radial velocity function phi_vel(m, u) metric = GeodesicBase.metric(m, u) inv_metric = inv(metric) E = my_energy(m.M, m.a, u) Lz = my_angmom(E, m.M, m.a, u) inv_metric[4, 4] * Lz - inv_metric[4, 1] * E end
This code snippet defines utility functions for calculating the initial radial velocity component for a given metric and position. The actual tracing code is a single function call:
using StaticArrays m = BoyerLindquist(M=1.0, a=0.0) u = [0.0, 6.0, π/2, 0.0] v = [0.0, 0.0, 0.0, phi_vel(m, u)] sol = tracegeodesics( m, u, v, (0.0, 1200.0), μ = 1.0, abstol = 1e-10, reltol = 1e-10, )
Tracing a couple of configurations and adding a boundary for the event horizon:
Note: for performance reasons, the integrator terminates at to avoid wasting time on doomed orbits.
Above are two orbits corresponding to circular and for and . The dashed line has an orbit at , whereas the solid line , i.e. an orbit just within the ISCO, and one exactly on the ISCO.
The integration is run for a total affine time of , as up until around both orbits are circular. In theory, the inner orbit is still above and should be able to remain a circular orbit, but due to the long integration times, we see the machine error contribute enough that the orbit becomes unstable, collapses and falls into the black hole. Other values of produce radically different unstable orbits, which all diverge in one way or another if the integrator is run for long enough.
We can investigate the stability of the integrator by setting the integration time to , and tracing all possible circular orbits for a specific and . A measure of deviation we employ is designed to tell us if the orbit collapsed inwards or outwards, and is calculated
such that we can distinguish three cases:
implies the orbit is stable and remained at the same radius,
; the orbit collapsed and spiraled inwards (as in the above figure), and
; the orbit spiralled outwards.
Note that the final two cases are not sufficient conditions for the orbit to fall into or completely escape the black hole respectively. This measure instead tells us the majority of the orbit was either within or beyond the initial radius, but some orbits may have crossed back and forth.
This figure is for and , since the effect of is only to translate the distribution left and right, and this configuration illustrates the regimes clearly. The radii considered are in steps of , and the pattern of is highly dependent on this, indicating that whether the orbit collapses or expands is due to machine precision. A few interesting things to note; the integrator is able to hold stable orbits all the way until the ISCO for long integration times. Between the marginally bound radius and the ISCO, deviations from the orbit are relatively small, which would indicate that the orbit remains circular for most of the integration time, before diverging – which is what we would expect from hyperbolic energies when dealing with numeric simulations. Between the photon orbit and the marginally bound orbit is where we see orbits that either collapse or expand extremely far away.
Within the photon horizon, the integration terminates after one or two steps towards the event horizon, which is why we gradually see approach unity, as only a single point remains in the integration result.
Particularly pronounced are the different boundaries, causing clearly different general behavior for the integrator. To investigate this further, we'll vary and introduce a measure of stability, coined as simply the normalized squared sum of the residuals in , expressed
chosen to not penalize greater radii orbits for small fluctuations – hence divide by initial radius – and to not penalize orbits with many steps by dividing by total integrated points . The motivation for using over is to be able to clearly differentiate regions where circular orbits are possible without penalizing eventual divergence in the same way does.
Applying this measure for a range of and :
Here, have been coloured transparent.
Overlayed are the different orbit boundaries. These boundaries align well with the different integration regimes, which is always nice to see. There are however a few anomalies, namely those orbits close to the event horizon for highly retrograde black holes, , where we see very poor stability, as the geodesics are launched out of the system from small energetic perturbations.
The small region of white within the event horizon in the bottom left corner is due to the integration being able to take a single step before it realizes any violation and terminates; it is, of course, a pure pathology.
Up until this point we have been able to use SageMath to derive results for and for circular orbits which hold all the way until . We have also confirmed that these values lead to circular orbits, and that the stability (and presence) of these orbits aligns with the different special boundaries. But what if we don't have good solutions for and ?
Generalizing the last section; can we reliably and quickly find circular orbits without any a priori knowledge of the velocity vectors, other than that we are searching in the equatorial plane?
The temptation here is to brute-force search over the parameter space until some stability condition is met, but we can be more elegant with this by using Optim.jl, which optimizes (i.e. finds the minima of) univariate functions. Optim.jl searches over a bounded range of any single argument function using the golden-section search algorithm.
For our purposes, the implementation needs a function which describes the degree of circularity of our orbits, which is effectively just from eq. (29).
The boilerplate is a few utility functions:
function trace_single_geodesic(m, u, vϕ) v = [0.0, 0.0, 0.0, vϕ] tracegeodesics( m, u, v, # use a fairly long, but not excessively long integration time (0.0, 1000.0), μ=1.0, abstol=1e-10, reltol=1e-10 ) end Qs(rs) = sqrt(sum((rs ./ rs .- 1.0).^2) / length(rs)) # utility function to trace and evaluate some trial `vϕ` function estimate_stability(m, u, vϕ) sol = trace_single_geodesic(m, u, vϕ) rs = selectdim(sol, 1, 6) Qs(rs) end
Now we come to the crux, which is wrapping
estimate_stability into a univariate function for Optim.jl. This is achieved through
using Optim function find_vϕ_for_orbit(m, r_init; lower_bound=0.0, upper_bound=0.1) u = [0.0, r_init, deg2rad(90.0), 0.0] res = optimize( vϕ -> estimate_stability(m, u, vϕ), lower_bound, upper_bound, GoldenSection() ) Optim.minimizer(res) end
The limitation of the univariate optimizer is the bounding window of possible values; estimating the bounding box too low risks excluding the true optimum, and estimating the bounding box too large risks local minima and bad high iteration count, which in turns leads to poor convergence. To try and mitigate this, we'll use a rolling window that continuously rescales these boundaries after each discovered orbit.
We can bake an assumption into this rolling window, which is that stable orbits closer to the singularity are more likely to have a higher initial than those further out, and solve the system backwards, starting with a conservative flat-space estimate for the velocity, and progressively work inwards towards the black hole. Half of this assumption is pretty robust, since the metrics we are dealing with are asymptotically flat, but it may not always be the case that orbits closer to the singularity rotate faster.
In code, we express this rolling window as
function find_vϕ_for_orbit_range(m, rs; lower=0.0, upper=0.1) map(reverse(rs)) do r vϕ = find_vϕ_for_orbit(m, r; lower_bound=lower, upper_bound=upper) # continuously adjust based on heuristic lower = vϕ * 0.9 upper = vϕ * 2.0 vϕ # call reverse twice so the order is maintained end |> reverse end
We can then solve for circular orbits over a range of with
m = BoyerLindquist(M=1.0, a=-0.4) vϕ = find_vϕ_for_orbit(m, 10.0) r_range = 1.1:0.1:10.0 vϕs = find_vϕ_for_orbit_range(m, r_range)
The solver takes on average 35 iterations (36 function calls) to converge to a minima of per orbit. Shorter integration times will also reliably converge in approximately the same number of function calls, with the routine being able to estimate stability after as little as a quarter of a full orbit reliably, up until approximately the ISCO. Within the ISCO, higher integration times are needed to accurately find solutions which illustrate the different orbital boundaries.
Calculating the energy for each orbit and plotting these again eq. (19) shows very good agreement to within for numerical accuracy:
Of course, the region within the photon orbit is impossible to probe with this technique, since the integrator cannot handle this domain. The negative energy regimes are therefore also impossible to discover with this method. Nevertheless, it proves reliable for finding stable orbits, up to the initial domain of the bounding window.
I'd also note here that this prototype code is able to discover those 100 stable orbits in just under a second, with approximately O() scaling. Some aspects of this method could be optimized, but that is beyond the intention of this blog post.
We can get good approximations of analytic solutions using both SageMath and integrator-driven automated circular orbit discovery. Both methods are faced with limitations, however, in how they (fail to) probe within the photon radius.
In the case of the SageMath, the negative energy description of the Kerr metric is entirely lost, and the angular momenta within the photon radius are inaccurate at best. When using the integrator, we cannot even probe within this radius reliably. It also has a practical limitation; SageMath is an unreliable solver at the best of times. Often the expressions it produces when invoking its
solve function are a mess, if it is able to solve them at all. I do not know how well this work will apply to metrics more complex than Kerr (which is the intended purpose after all). Though fortunately it seems that some fairly generic operations can go some distance to remedying this.
SageMath merits further investigation, and I will make the relevant SageMath notebooks for recreating the analysis in this blog available at some point.
The integrator suffers from an initial guess problem for the rolling window bounds on the possible range of , but other than that is able to work without issue directly from the input metric. It is also able to describe enough of the space of circular orbits that the different orbital boundaries are visible, and will no doubt prove to be useful when studying other metrics.
Overall, I am pleased with how well the integrator performs for such orbits, and am confident these methods should be applicable to other spacetimes too.
The next step is to apply this work to general static, axis-symmetric spacetimes and modified Kerr metrics, and to assemble a full pipeline for either deriving and wrapping symbolic expressions into Julia functions, or a pipeline for automatic orbit discovery in the integrator ecosystem. This will then be used to calculate the redshift values for geometrically thin discs in the equatorial plane, to move the integrator in the direction of observational results purely from metrics.
Another avenue to later explore are the class of spherical orbits, which are not constrained to the equatorial plane, but nevertheless are able to maintain a constant radius.
A future post may also investigate ways of creating or even machine learning angular velocity tables for circular orbits, and how these could be shared and use in integration models.
|||J M Bardeen, W H Press, S A Teukolsky (1972). Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radiation. APJ, 178:347-169.|
|||C T Cunningham (1975). The effects of redshifts and focusing on the spectrum of an accretion disk around a Kerr black hole. APJ, 202:788-802.|
|||B Carter (1968). Global structure of the Kerr family of gravitational fields. PHYS REV 174:5.|