SCIENCE · INVESTIGATE AND EXPLAIN

Reconstruct a lens without mistaking images for sources

Gravity changes the paths of light, so a foreground mass can create multiple apparent images of one background source. Lensing depends on projected mass and the observer–lens–source geometry. It provides a mass-sensitive observation different from the source’s own luminosity or a tracer’s orbital motion.

Learning outcome: Solve both signed roots and verify the original equation for each. Reconstruct sum/product before explaining what a more complex lens geometry would make uncertain.

Starting recap

Keep galaxy mass, source calibration, expansion history and independent cosmological evidence distinct.

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

At M=2 and β=0.4, image positions are approximately 1.62829 and−1.22829 angle units. Their sum is 0.4 and minus their product is 2. The separation is about 2.85657, not simply the source offset or twice the offset.

Open the experiment

Predict

Do two gravitational images necessarily imply two separate luminous sources?

Your prediction and reason

Experiment

Solve both signed roots and verify the original equation for each. Reconstruct sum/product before explaining what a more complex lens geometry would make uncertain.

Reconstruct two images under a stated lens geometry

Worked teaching model · synthetic values · Original thin point lens with fixed distance factor scaled so Einstein angle squared equals mass. Signed lens equation beta=theta−M/theta. Two images are angular positions, not two sources; no extended lens, microlensing light curve or unique real halo inference.

Normalized point-lens mass: 2 mass units Source angular offset: 0.4 angle units

Reset keeps your prediction, captures and practice history.

Two signed image positions can constrain a point-lens model but are not independent physical sources.

Original thin point lens with fixed distance factor scaled so Einstein angle squared equals mass. Signed lens equation beta=theta−M/theta. Two images are angular positions, not two sources; no extended lens, microlensing light curve or unique real halo inference.

Original conditional model ledger
Quantity	Value	Unit
Einstein angle	1.414214	angle units
Positive-side image position	1.628286	signed angle units
Negative-side image position	-1.228286	signed angle units
Source offset reconstructed from image sum	0.4000000	angle units
Mass reconstructed from minus image product	2.000000	mass units
Image separation	2.856571	angle units

These supplied models are not current measured cosmological parameters, a new galaxy discovery or independent actual observing evidence.

Two signed image positions can constrain a point-lens model but are not independent physical sources.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

1. Define the cosmic evidence

2. Connect mechanisms and alternatives

The original thin point-lens geometry uses normalized angular units in which Einstein-angle squared equals lens mass M. The signed one-dimensional lens equation is β=θ−M/θ. Its two roots are[β±sqrt(β²+4M)]/2; one lies on either side of the lens for the positive source offsets supplied.

The image-position sum equals β and their product equals−M in this normalized geometry. These identities reconstruct the two model parameters from the two signed positions. The negative image coordinate indicates the opposite angular side, not negative physical mass or a second negative-distance source.

Real galaxy lenses have extended mass, external shear, uncertain distances and possible degeneracies between source and lens models. Multiple images alone do not fix an arbitrary halo profile, and a point-lens inversion cannot be silently applied to a complex arc. Repeated source structure, time delays and independent distance/dynamical evidence can test the interpretation.

3. Work the supplied comparison

4. Predict and investigate

Do two gravitational images necessarily imply two separate luminous sources?

Solve both signed roots and verify the original equation for each. Reconstruct sum/product before explaining what a more complex lens geometry would make uncertain.

Apply the evidence to a changed case

Use M=1 and β=0.2. Verify the image sum and product without rounding first.

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

The roots are(0.2±sqrt(4.04))/2. Their sum is 0.2 and product−1 exactly under the model; minus the product recovers the normalized mass. A real mass would also require the geometric scale.

Source and boundaries

Primary source. Checked 31 October 2026. Original model records are not new galaxy observations, current parameter measurements or a full cosmological inference pipeline. No physical observing procedure is requested.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Use M=1 and β=0.2. Verify the image sum and product without rounding first.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Distinguish a luminous population from most of a galaxy’s mass Next: Propagate a correlated calibration through a distance ladder

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