Dark matter dynamical friction on binary stars in the faintest dwarf galaxies: I. Analytical approach

The gist

Binary stars moving through a uniform dark matter background experience dynamical friction, which can significantly alter their orbital evolution and potentially enhance biases in velocity dispersion measurements in ultra-faint dwarf galaxies.

Dynamical Friction Framework

The dynamical friction exerted on a star of mass mi moving with velocity vi through a DM background of uniform density ρdm is described by the Chandrasekhar formula (12). The acceleration due to dynamical friction reads fi = −4πG2ρdmmi ln Λ A(X) vi/σ v 3i, where A(X) is a dimensionless monotonic function (1). The Coulomb logarithm ln Λ has been assigned the value of 10 for the remainder of this work (2). The mass of this wake is proportional to the stellar mass, meaning more massive stars generate larger wakes which slow them down more effectively (2). In a binary stellar system, the effect on internal orbital motion is given by fdf = −4πG2ρdm ln Λ × m1A(v1/σ) v1/v 3i − m2A(v2/σ) v2/v 3i (2).

Evolution Regimes for Binaries

The secular evolution of orbital parameters is analyzed in two opposite regimes: wide binaries and close binaries (3). The transition between these regimes occurs for binaries with typical separation a ∼ GM/vcm squared ∼ 100 AU / M M⊙ 3 km/s vcm squared (5). For wide binaries, where the orbital velocity is small compared to the CoM velocity, the orbital evolution depends on the relative orientation between vcm and x˙ (6). The energy evolution is dominated by first-order corrections because the zeroth-order part of Eq. (6) depends only on vcm and leads to a term proportional to ⟨x˙⟩ which vanishes upon time averaging (8).

Wide Binary Evolution

The evolution of the semi-major axis can be derived from its relation to orbital energy, ˙a = −aE/E ˙, and can be either negative or positive (10). For binaries with an orbit perpendicular to the CoM velocity (i = 0), the semi-major axis evolves as a scale-invariant function a(t) = a0 exp(−t/τw,1) (10). The wide binary energy loss timescale is defined as τw,1 = 16πG2ρdmµ ln Λ v 3cm A(vcm/σ)/−1 ∼ 5 × 109 yr 100 GeV/cm3 ρdm max(vcm, σ)3 km/s cubed (9). The evolution of the eccentricity and other orbital elements is characterized by equations derived from the orbit-averaged evolution (11).

Close Binary Evolution

For tight binaries, where the internal orbital velocity satisfies ˙x ∼ pGM/a ≫ vcm ∼ σ, the leading order of Eq. (2) is fdf = −4πG2M3µρdm ln Λ x˙ cubed × [1/m1A(m1x˙ Mσ + 1/m3A(m2x˙ Mσ x˙.)] (13). In this limit, the friction force is confined to the orbital plane, allowing one to set i = 0 (i.e., take the orbital plane to coincide with the xy-plane) (13). The close binary evolution timescale is defined as τc = 8πρdm ln Λ√GM 3/2µ / (1/m3A(m1+m2)a 3/2) −1 ∼ 10 12 yr 100 GeV/cm3 ρdm AU a0 3/2 (15). In the case of circular orbits, the semi-major axis evolves as a(t) = a0 (1 + 3t/2τc)−2/3 (16).

Effect on Stellar Populations

When applied to a toy model of binary stars in environments typical of ultra-faint dwarf galaxies, the results show that binaries can undergo strong eccentricity oscillations on Gyr timescales while their semi-major axes evolve more gradually (6). For tight binaries, eccentricity tends to increase and the semi-major axis decreases, although on a timescale much larger than the age of the Universe (6). This evolution typically leads to a larger fraction of high-eccentricity systems and an overall rise in the orbital velocities of the stellar population (6). Furthermore, there is an increase in small periastron passages, with approximately 10% of binaries reaching sub-AU separations at some point due to dark matter friction over the age of the galaxy (6).

Future Directions

Several improvements remain for future work, including clarifying the applicability of the Chandrasekhar formula to close binaries and characterizing dynamical friction in such systems using dedicated simulations that account for overlapping wakes (6). Additional physical effects such as galactic tides, three-body encounters, and migration toward the galactic centre driven by DM friction should be incorporated into comprehensive numerical integrations (6). The paper also notes that galactic tidal forces may dominate over internal DM friction in binaries wider than 2000 AU (15).

How it works

The analysis involves deriving analytical expressions and timescales for the secular evolution of orbital parameters for both wide and close binaries (Introduction). The general framework for dynamical friction acting on binary stars is introduced in Section II, using the Chandrasekhar formula to describe the friction exerted on a star in a DM background (2). The effect of dynamical friction on internal orbital motion is quantified by fdf = f1 − f2 (2), where f1 and f2 are the accelerations due to dynamical friction on each component.

Key Findings

1. Wide binaries undergo significant eccentricity oscillations, while their semi-major axes decrease more gradually (6).

2. Close binaries exhibit an increase in eccentricity and a decrease in semi-major axis, although this occurs on a timescale much larger than the age of the Universe (6).

3. The effect of dynamical friction can enhance biases from unresolved binaries in velocity dispersion measurements, which leads to an overestimation of the DM density in these systems (6).

4. The evolution timescales for wide binaries are comparable to or shorter than those for galactic tides depending on the separation, with galactic tides dominating over internal DM friction for wider separations (15).

5. For a population of 1000 low-mass binaries evolved for 13 Gyr under dark matter dynamical friction, there is an overall increase in the number of high-eccentricity pairs and a rise in the number of binaries with semi-major axes around 10 AU (6).

Timescale Comparison

A comparison of various timescales shows that galactic tides may dominate over internal DM friction for binaries wider than 2000 AU (15). Stellar encounters and dynamical friction on the centre-of-mass have timescales longer than τw,1 and τw,0 (B3). The sinking timescale of the binary pair as a whole remains shorter than the age of the Universe by a factor of a few (B3).

Conclusion

Binary stars can be significantly affected by dynamical friction in dense UFDs, potentially serving as probes for dark matter content in faint galaxies (Introduction). The study provides analytical formulas and timescales for the secular evolution of orbital parameters induced by DM dynamical friction (Introduction). The results indicate that binaries may undergo strong eccentricity oscillations on Gyr timescales in wide binaries, while tight binaries experience eccentricity growth and semi-major axis decrease (6). This evolution leads to enhanced biases on velocity dispersion measurements and strengthened tidal interactions between binary members (6).

How it works

The paper uses a simple binary star population as a toy model, drawing masses from a Chabrier initial mass function and initial eccentricities from the thermal distribution (V) The numerical solution for 1000 pairs evolved for 13 Gyr under dark matter dynamical friction is presented in Figure 3 (6).

Key Results Summary

The study successfully derives analytical expressions and timescales for both wide and close binaries, applying them to a simple binary star population in an environment typical of UFDs (Introduction). The results indicate that binary stars can be significantly affected by dynamical friction in dense UFDs, potentially enhancing biases in velocity dispersion measurements (Introduction). The paper concludes that binaries themselves could serve as a probe of dark matter in some of the faintest galaxies (Introduction).

Final Assessment

The study provides analytical formulas and timescales for the secular evolution of orbital parameters induced by DM dynamical friction (Introduction). The results indicate that binary stars can be significantly affected by dynamical friction in dense UFDs, potentially enhancing biases in velocity dispersion measurements (Introduction). The paper concludes that binaries themselves could serve as a probe of dark matter in some of the faintest galaxies (Introduction).

Summary

Binary stars moving through a uniform dark matter background experience dynamical friction, which can significantly alter their orbital evolution and potentially enhance biases in velocity dispersion measurements in ultra-faint dwarf galaxies. The study derives analytical formulas and timescales for the secular evolution of orbital parameters for both wide and close binaries, finding that wide binaries may undergo strong eccentricity oscillations on Gyr timescales while their semi-major axes evolve more gradually, whereas tight binaries experience eccentricity growth and semi-major axis decrease (6). This evolution leads to enhanced biases on velocity dispersion measurements and strengthened tidal interactions between binary members (6). The paper concludes that binaries themselves could serve as a probe of dark matter in some of the faintest galaxies (Introduction).

References

[1] M. Cirelli, A. Strumia, and J. Zupan, arXiv preprint, arXiv:2406.

1. textbf Enhanced Dark Matter Content Estimation Bias Correction for Ultra-Faint Dwarf Galaxies (UFDs) in Velocity Dispersion Measurements: The improved AI can now calculate a correction factor for velocity dispersion measurements based on orbital evolution, explicitly addressing the effect of dynamical friction on unresolved binaries, as suggested by the paper: dynamical friction, notably, can enhance the bias from unresolved binaries in velocity dispersion measurements.

2. textbf Predictive Modeling of Binary Orbital Evolution Timescales: The improved AI can derive analytical expressions and timescales for secular evolution for both wide and close binaries in uniform dark matter backgrounds, allowing it to predict how orbital parameters like semi-major axis decrease more gradually or how eccentricity oscillates based on the binary type.

3. textbf Multi-Scale Dynamical Friction Simulation: The improved AI can integrate the general equation of binary evolution (Eq. 2) for populations of stars, using simplified equations for wide (tight) binaries when appropriate, to simulate the long-term effects over timescales relevant to UFDs, such as 13 Gyr.

4. textbf Identification of High-Eccentricity and Small Periastron Systems: The improved AI can identify the resulting population changes after 13 Gyr, specifically observing an overall increase in the number of high-eccentricity pairs and a rise in the number of binaries with semi-major axes around ∼ 10 AU, and predicting that approximately 10% of the binaries in the sample therefore reach sub-AU separations at some point due to dark matter friction.

5. textbf Constraint Generation for Dark Matter Density: The improved AI can utilize the derived evolution timescales, comparing them with other relevant mechanisms like galactic tides and stellar encounters, to constrain dark matter content by comparing observed orbital evolution against theoretical predictions, such as assessing when galactic tides may dominate over internal DM friction in binaries wider than ∼ 2000 AU.

We study binary stars moving through a uniform dark matter background and experiencing dynamical friction. The centre-of-mass motion of the pairs is taken into account. We derive formulas and timescales for the secular evolution of the orbital parameters for both wide and close binaries. Applying these results to environments typical of the faintest ultra-faint dwarf galaxies, we show that some binaries undergo significant eccentricity oscillations while their semi-major axes gradually decrease over a Hubble time. At the population level, dynamical friction depletes wide binaries and increases the fraction of highly eccentric systems. Importantly, the effect is not strongest at the highest dark matter densities, since the associated increase in velocity dispersion weakens friction. With future, more detailed theoretical studies and improving observational capabilities, binary stars may serve as a tool to probe the dark matter content of the faintest galaxies.