Chandrasekhar Dynamical Friction Timescale Calculator
Overview
The Chandrasekhar dynamical friction formula quantifies how a massive object slows down as it moves through a background of stars, gas, or dark matter. This calculator estimates an orbital decay timescale and the distance travelled before significant slowing, using the standard Chandrasekhar prescription.
Typical applications include satellite galaxies orbiting larger hosts, globular clusters spiralling toward galactic centres, or massive black holes moving through stellar backgrounds. By adjusting the mass, background density, velocity, and Coulomb logarithm, you can explore how quickly different systems lose orbital energy.
Chandrasekhar Dynamical Friction Formula
In a homogeneous, isotropic background with a Maxwellian velocity distribution, the magnitude of the deceleration due to dynamical friction can be recast as a characteristic decay time. A commonly used approximate timescale is
where:
- t is the characteristic dynamical friction timescale.
- v is the speed of the massive body relative to the background.
- G is the gravitational constant.
- M is the mass of the moving body.
- ρ is the mass density of the background medium.
- lnΛ (often written lnΛ) is the Coulomb logarithm, capturing the effective range of impact parameters that contribute to the drag.
The calculator evaluates this expression using your inputs, after converting them to SI units, and then expresses the resulting timescale in years. It also reports the distance the object would travel at its initial speed over that time, highlighting how far the body moves before friction substantially alters its orbit.
Units and Conversions
The input units are chosen to match common astrophysical practice:
- Mass M: solar masses (M☉).
- Background density ρ: solar masses per cubic parsec (M☉/pc3).
- Velocity v: kilometres per second (km/s).
- Coulomb logarithm lnΛ: dimensionless.
Internally, the script converts these to SI units by applying:
- 1 M☉ ≈ 1.9885 × 1030 kg.
- 1 pc ≈ 3.0857 × 1016 m, so 1 pc3 ≈ (3.0857 × 1016 m)3.
- 1 km/s = 103 m/s.
The gravitational constant is taken as G ≈ 6.6743 × 10−11 m3 kg−1 s−2. After computing the timescale in seconds, the calculator converts it to years using 1 year ≈ 3.154 × 107 s. The travelled distance is obtained from
and then reported in convenient astrophysical units (for example, kiloparsecs) for easier interpretation.
Interpreting the Results
The output timescale corresponds to the order of magnitude time over which dynamical friction significantly alters the orbit of the massive object. It is not an exact merger time, but rather a guide to how quickly orbital energy is drained.
Some broad regimes:
- t much shorter than 1 Gyr: the body experiences rapid orbital decay. Massive clusters or satellites will spiral in on timescales short compared with galactic evolution, and their current configuration is likely transient.
- t of order a few Gyr: decay is important over a Hubble time, but objects may complete many orbits before merging. This regime is common for satellite galaxies in Milky Way–like haloes.
- t comparable to or longer than the age of the Universe (≈ 13.8 Gyr): dynamical friction is weak. The orbit is effectively stable against decay on cosmological timescales, and other processes may dominate the evolution.
The travel distance complements the timescale. If the travelled distance is several times the current orbital radius, the object can circle its host many times before friction causes substantial inward migration. If the distance is comparable to or smaller than the current radius, notable orbital decay occurs within roughly one orbital period.
Worked Example
Consider a satellite galaxy orbiting a Milky Way–like halo. Suppose:
- Mass M = 108 M☉.
- Background density ρ = 0.1 M☉/pc3 (typical of an inner halo or dense region).
- Velocity v = 200 km/s.
- Coulomb logarithm lnΛ = 10.
Entering these values into the calculator yields a characteristic decay time on the order of billions of years and a travelled distance of many tens of kiloparsecs. This implies that while dynamical friction is important over the lifetime of the galaxy, the satellite completes many orbits before its energy loss leads to a final merger.
You can experiment by increasing M or ρ to see how the timescale shortens. Doubling the mass or local density roughly halves the decay time, whereas doubling the velocity increases the timescale by a factor of about eight, because the friction scales with v−3.
Comparison with Other Timescales
The dynamical friction timescale should be interpreted alongside other relevant dynamical times. The table below sketches qualitative differences.
| Timescale | What it Measures | Typical Use |
|---|---|---|
| Dynamical friction timescale | Time for a massive body to lose orbital energy and angular momentum due to interactions with a background medium. | Satellite galaxy or globular cluster orbital decay; massive black hole sinking toward a galactic centre. |
| Orbital period | Time for one orbit at the current radius and velocity. | Short-term orbital evolution, phase-mixing, and resonance analysis. |
| Crossing time / dynamical time | Time for a typical particle to traverse the system once. | Stability of clusters and galaxies; virialization and relaxation processes. |
| Two-body relaxation time | Time for cumulative small-angle encounters to significantly change stellar velocities. | Long-term evolution of star clusters and dense stellar systems. |
Assumptions and Limitations
The Chandrasekhar formula is powerful but idealised. Results from this calculator should be viewed as order-of-magnitude estimates rather than precise predictions. Key assumptions include:
- Homogeneous, isotropic background: the method assumes the massive body moves through a smooth medium with constant density and no strong gradients. Real galaxies have disks, bulges, and haloes with varying densities.
- Maxwellian velocity distribution: background particles are assumed to follow a Maxwellian distribution. Deviations, such as cold streams or coherent substructures, can change the effective friction.
- Weak, cumulative encounters: the Coulomb logarithm approximation is based on many small-angle gravitational encounters. Rare, strong encounters with comparably massive bodies are not explicitly treated.
- Straight-line or gently curving motion: the formula is most accurate when the orbit does not change direction too rapidly on the scale of the interaction region. Highly eccentric orbits may require more detailed modelling.
- Single-component background: the calculation uses a single density ρ. Real systems can contain multiple components (stars, gas, dark matter) with different densities and velocity dispersions, each contributing differently to friction.
Because of these idealisations, you should treat the computed timescale as a guide to whether dynamical friction is important and how it scales with parameters, rather than as a replacement for detailed N-body simulations or orbit integrations in complex potentials.
Practical Usage Tips
When choosing input values:
- Use masses from about 105–1010 M☉ for typical dwarf galaxies, globular clusters, or massive black holes.
- Adopt densities between roughly 10−4 and 0.1 M☉/pc3 for diffuse haloes, and up to a few M☉/pc3 for dense inner regions.
- Set velocities in the range 50–300 km/s for orbits in Milky Way–like galaxies, or higher for galaxy clusters.
- Choose lnΛ between about 5 and 15; using 10 is a reasonable default when detailed modelling is unavailable.
Students can use the tool to see how strongly the timescale depends on each parameter by varying one input at a time. Researchers can obtain quick back-of-the-envelope estimates before running more computationally expensive simulations.