Astronomy · Comparing galaxies

Interactions, mergers and changing structure

Calculate the difference between physical attraction and a relative tidal acceleration, then connect that distinction to the evidence and limits of galaxy-interaction histories.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

Back to the chapter

After reading, you can

Distinguish a common gravitational acceleration from a spatial acceleration gradient.
Reproduce exact and small-separation relative accelerations in a declared point-perturber model.
Explain why a flyby, a merger and an observed distorted image do not imply identical histories.

An interaction changes more than the outline

Galaxies exert gravity on one another. An encounter can alter stellar orbits, move gas and redistribute angular momentum without requiring most individual stars to collide directly. The large spaces between stars make direct stellar collisions a different process from an encounter between entire galaxies.

Gas can also shock, radiate and dissipate orbital energy into other forms. Stars, gas and the surrounding gravitating distribution therefore need not respond identically. Features such as bridges, tails or disturbed gas provide clues to an encounter, but their interpretation depends on the viewing direction and physical model.

A flyby and a merger are different possible outcomes. Whether two systems become one depends on their orbit, mass distribution and exchanges of energy and angular momentum. A close-looking pair in a projected image does not by itself establish their true separation, a future merger or a merger time.

To isolate one ingredient, begin with a spatial gradient of gravity at a single instant. The following model is an original teaching calculation. It does not evolve two extended galaxies or predict whether any real stellar system is disrupted.

All three physical attractions point toward the perturber

Use a Newtonian weak-field model with a point perturber of mass M_p at coordinate X = D > 0. A reference body’s centre is at X = 0. Compare test positions at the near side X = +r and far side X = −r, where r = xD and 0 < x < 1. Let G be the gravitational constant.

Define the positive acceleration scale A0 = GM_p/D². The perturber’s physical acceleration at the centre is +A0. At the near point it is +A0/(1 − x)², and at the far point it is +A0/(1 + x)². All three accelerations point in the positive X direction toward the same perturber.

The near point is attracted more strongly than the centre because its distance is D − r. The far point is attracted less strongly because its distance is D + r. There is no reversed physical attraction on the far side in these equations.

The equations supply only the instantaneous acceleration due to the external point perturber along this line. The body’s internal gravity, transverse forces, motion of the perturber and subsequent orbital evolution are not specified. Do not interpret the supplied terms as each test position’s complete acceleration.

Relative acceleration can have the opposite sign

Relative to the accelerating centre, subtract the centre’s +A0 from each external acceleration. The near-side relative term is A0[(1 − x)⁻² − 1], which is positive. The far-side relative term is A0[(1 + x)⁻² − 1], which is negative. The negative sign means that the far point falls toward the perturber less rapidly than the centre.

At x = 0.10, the physical near acceleration is (100/81)A0 and the physical far acceleration is (100/121)A0. Subtracting the central A0 gives near +(19/81)A0 ≈ +0.234567901A0 and far −(21/121)A0 ≈ −0.173553719A0.

In this centre-relative description, the two sides tend to separate along the line toward the perturber. That statement describes an external differential forcing. Whether a self-gravitating body actually stretches, oscillates, remains bound or loses material also depends on its internal dynamics and on the duration and trajectory of the encounter.

A diagram must therefore label the frame and arrow scales. A negative relative arrow on the far side must never be read as repulsion from the perturber. The exact table also shows that finite-separation near and far terms have unequal magnitudes.

Original instantaneous point-perturber calculation. Physical accelerations are normalized to A0 = GM_p/D² and are all positive X. Relative terms subtract the centre’s +1. Linear terms are small-x approximations, not exact results at the listed finite separations.
x = r/D	Physical near / A0	Physical far / A0	Relative near / A0	Relative far / A0	Linear near / far
0.05	1.108033241	0.907029478	+0.108033241	−0.092970522	+0.10 / −0.10
0.10	1.234567901	0.826446281	+0.234567901	−0.173553719	+0.20 / −0.20
0.20	1.562500000	0.694444444	+0.562500000	−0.305555556	+0.40 / −0.40

The small-separation approximation has a domain

Expanding (1 ∓ x)⁻² to first order for x much smaller than one gives relative terms approximately ±2xA0 = ±2GM_p r/D³. The common acceleration scales as D⁻², while the leading relative gradient across a fixed small r scales as D⁻³.

At x = 0.20, the first-order terms would be ±0.40A0, compared with exact +0.5625A0 and −0.305555556A0. The linearization erases the finite-separation asymmetry. Writing an approximation sign is essential; the first-order expression is not an exact replacement at every r/D.

The point-perturber calculation also does not describe all directions of a full tidal field. It isolates the along-line comparison and a chosen instantaneous geometry. Real extended mass distributions, close overlap and time-dependent orbits require a more complete gravitational model.

Bovy’s treatment of tides connects relative forcing to gradients in a gravitational field. The useful distinction here is between a common acceleration, which moves the chosen centre too, and a difference in acceleration across a system. Neither the exact one-dimensional calculation nor its linear form supplies a galaxy merger rate or a disruption criterion.

Attraction and relative acceleration

Scroll sideways to see the full diagram. Its explanation is below.

Original instantaneous one-dimensional Newtonian point-perturber model, not observed galaxy data or an encounter simulation. A point perturber P is at X = D, the body centre C is at X = 0, the near test location N is at X = plus r, and the far test location F is at X = minus r. D is positive and r/D = 0.10 in the drawing. All rows share these same X positions; their vertical separation is only page layout. The position axis X/D uses one common linear scale. Define A0 = G Mp / D squared, the positive perturbing acceleration at C. All physical perturbing accelerations point towards P, in the plus-X direction. Physical accelerations g/A0 are 100/121 = 0.826446280992 at F, 1 at C, and 100/81 = 1.234567901235 at N. Following the centre means subtracting g at C from the perturbing acceleration at each location. The external relative contributions Delta g/A0 are minus 21/121 = minus 0.173553719008 at F, zero at C, and plus 19/81 = plus 0.234567901235 at N. The far-side relative arrow points away from P because its physical acceleration towards P is weaker than the centre acceleration. The relative arrows use an explicitly declared ten-times-longer drawing scale: the blue physical scale bar represents 1 A0, and the equally long orange relative scale bar represents 0.1 A0. Within each panel all arrows share that panel scale. The centre has zero relative contribution and no relative arrow. The fallback also supplies exact physical and centre-subtracted values for r/D = 0.05, 0.10 and 0.20 and contrasts them with the first-order approximation plus or minus 2r/D; the approximation is not the exact finite-r formula. This diagram shows only the external perturbing contribution. Internal self-gravity, gas physics, trajectories, disruption and merger outcomes are omitted.

A changing galaxy needs more than one snapshot

During an encounter, orbital energy and angular momentum can be redistributed among different components. Dynamical friction describes a collective gravitational exchange with the surrounding distribution; it does not mean that total energy simply vanishes. Gas can additionally radiate energy after heating or shocks.

Interactions can rearrange material and sometimes enhance star formation when the gas conditions permit it. They do not guarantee a starburst in every case. Nor does one simplified merger picture establish that every elliptical galaxy formed through an identical sequence of events.

Images, kinematics, gas distributions and stellar-population measurements constrain possible histories together. Simulations can test whether a proposed orbit and mass model produce compatible features. A visually plausible match is conditional evidence, rather than proof that the inferred past is unique.

The final lesson in this chapter compares selected galaxy samples. It will keep actual differences in the stipulated parent populations separate from detection effects and from a comparison made after matching population composition.

Worked examples

Explain the negative relative far-side arrow

At r/D = 0.10, why is the far-side relative acceleration negative when gravity still attracts it toward the perturber?

The far point has physical acceleration +(100/121)A0 toward the positive-X perturber.
The reference centre has the larger physical acceleration +A0.
Subtracting the centre gives (100/121 − 1)A0 = −(21/121)A0.

The negative centre-relative term measures weaker attraction than the centre’s, not physical repulsion.

Check a linear approximation against the exact terms

At r/D = 0.20, does the first-order prediction give the exact near- and far-side terms?

The first-order terms are ±2xA0 = ±0.40A0.
The exact near term is [(0.8)⁻² − 1]A0 = +0.5625A0.
The exact far term is [(1.2)⁻² − 1]A0 ≈ −0.305555556A0.

No. The finite-separation terms differ from the linear values and are unequal in magnitude.

Try the reasoning

Does the far-side negative relative term mean the perturber physically repels the far point?
Show the answer and reasoning
No.
Its physical attraction remains positive; subtracting the centre’s larger acceleration gives a negative difference.
For fixed small r, do the common point-mass acceleration and its leading relative difference have the same D scaling?
Show the answer and reasoning
No.
They scale as D⁻² and D⁻³ respectively in the supplied model.
Do the external acceleration terms alone determine whether the body becomes unbound?
Show the answer and reasoning
No.
Its internal gravity and the time-dependent encounter are not specified.
Must every galaxy encounter become a merger with enhanced star formation?
Show the answer and reasoning
No.
The outcome depends on the orbit, energy and angular-momentum exchanges, gas conditions and other physical properties.

What to carry forward

An external attraction and a centre-relative tidal difference are different quantities.
The supplied exact finite-separation terms reveal the limits of a first-order tidal approximation.
Galaxy-interaction histories require combined evidence and dynamical models rather than one image or one force estimate.

References

Teaching resources and research supporting this lesson’s explanations and examples.