This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

### Session: New Approaches to Classical Dynamical Problems I

Marc Favata (Montclair State University) (invited)

#### Abstract

Solving the relativistic two-body problem is difficult. Motivated by the construction, operation, and recent upgrades of interferometric gravitational-wave detectors, significant progress on this problem has been achieved over the past two decades. I will provide a summary of techniques that have been developed to solve the relativistic two-body problem, with an emphasis on semi-analytic approaches, their relevance to gravitational-wave astronomy, and remaining unsolved issues.

#### Notes

- Gravitational wave (GW) detector networks:
- AdvLIGO/Virgo+ (~2015+)
- Upgrades complete as of 1 April 2015!
- ~3 yr to get to final design sensitivity
- Upgrade: ~10 times more sensitive

- Kagra (~2018)
- LIGO-India (~2022)
- Pulsar timing arrays (~now)
- NANOgrav, EPTA, PPTA

- Future: third-gen LIGO

- AdvLIGO/Virgo+ (~2015+)
- GW sources
- Merging stellar-mass compact-object binaries (NS or BH)
- measure masses and spins
- determine merger rates

- core-collapse SN
- isolated neutron stars
- cosmic strings, stochastic bg
- unexpected
- Low-freq sources (LISA):
- merging SMBHs
- extreme-mass ratio ???
- ???

- Merging stellar-mass compact-object binaries (NS or BH)
- Coalescing binaries
- phases: inspiral (periodic, long), merger (frequency chirp and peak amplitude, short), and ringdown (damping)
- During merger and ringdown, the two holes merge and the remnant undergoes damped oscillations

- Why two-body GR is hard
- Einstein’s eqs. are just a lot more complicated
- Newton: only mass density
- E: density, vel., kinetic energy, etc.
- Highly nonlinear

- Solutions to E equations
- Exact solutions: Kerr and FrW
- Perturbation theory: PN theory, BH pert. theory
- Numerical relativity: finite resolution, inexact ICs, cpu time

- Numerical Relativity
- Not really viable until ~2005, despite efforts from the 1960s
- Mergers now routine
- Future: detailed exploration of BH/BH param space
- NS+BH, NS+NS: realistic EOS, mag. fields, neutrinos…
- Computationally expensive beyond ~10 orbits
- NS+NS: 8000 orbits, NS+BH: 1800 orbits, BH+BH: 300 orbits
- Orbital and radiation-reaction timescales
- small mass ratios < 1/10
*very*costly - Current best achievement: 176 orbits

- Need for phase accuracy
- LIGO data is noisy $\rightarrow$ need good signal template
- integral of an oscillating function
- phase evol. of signal needs to be accurate to fraction of a cycle
- Templates: >10 parameters

- PN approx.
- write E eqs as perturbation on flat-space wave eqn
- series expansions
- plug expansions back into E eqs
- iterate
- gets very messy very quickly
- radiative effects important
- orbital phasing is where the information lives — need to get to as high an order as possible
- need to get to 3.5PN ($v^7$)

- high-order harmonics can be important
- “memory modes”: non-oscillatory but time-varying modes (secular effects)
- nonlinear effect
- GWs themselves produce GWs(!)

- Spin effects
- aligned: minor correction
- non-aligned: mess
- eqs to describe spin evolution must also be solved

- Eccentricity effects
- GWs damp eccentricity, so often ignored
- But eccentric signals possible from binaries
- periastron precession
- eccentricity-induced modulations to orbital phase & amplitude
- corrections also need to be high-order

- Tidal interactions
- near end of inspiral
- tidal distortionparameterized in terms of tidal Love number
- Measuring tidal Love number provides useful constraints

- types:
- electric
- magnetic
- shape

- electric Love number most observationally relevant

- BH pert. theory
- EMRI orbits
- very complicated — rich structure, resonant effects
- produces interesting “jumps” in phasing and orbital elements

- self-force approach

- Conclusion: $2^{nd}$ gen network of GW detectors is coming online now
- Need good modeling
- Need good control over systematic errors (hence high-order PN work)

Here you may find a simple post-Newtonian solution for Mercury’s orbit precession

Gravity is a little big bigger than in Newton’s law; it increases with speed -kinetic energy- where the maximum is the double gravity in the case of light.

Global Physics also predicts the anomalous precession of Mercury’s orbit as Paul Gerber did 20 years before Einstein. https://molwick.com/en/gravitation/077-mercury-orbit.html