Resolution, sensitivity and contrast are different

Ask which capability the observation needs

An instrument can collect a measurable signal from a source while leaving its shape unresolved. It can also distinguish two bright nearby sources yet struggle with a much fainter neighbour beside a bright one. These are different observing questions.

Angular resolution concerns distinguishable detail on the sky. Sensitivity here means the ability to measure a faint astronomical signal under specified band, exposure, background, response and analysis conditions. Contrast concerns a relative signal difference, often a faint companion beside a bright source.

Each term needs a context. Angular resolution can be described through a point-response model and a chosen criterion. A limiting flux needs an uncertainty or detection procedure. A contrast statement needs a ratio convention, angular separation and residual-light conditions.

The preceding instrument chapter introduced these properties as design choices. This chapter explains the physical limits and the distinctions needed to interpret those choices.

The recorded pattern can constrain more than its shape

A point-spread function, or PSF, is the image response to an ideal point source under a stated system model. A sufficiently small source can appear as that finite response rather than reveal its own physical outline. “Unresolved” describes that limitation on structure.

Its total extracted flux and a model-based position can still be measured, with appropriate calibration and uncertainty. Measuring the location of a point-response centre is a different task from resolving the source’s surface or separating an arbitrary pair.

An apparent single response may also contain several unresolved contributions. Without additional information, the total does not uniquely allocate flux among them. The earlier multiwavelength two-cell example illustrated that loss of allocation information.

Use the measurement that is supported. A useful flux estimate does not require claiming a resolved shape; an unresolved shape does not mean the instrument recorded no information.

A faint companion is measured against a bright-source environment

For this lesson define contrast as companion band flux divided by star band flux, measured in the same band and compatible units. Some contexts use a reciprocal or another convention, so state the chosen definition. A smaller companion-to-star ratio means a fainter relative companion.

In a synthetic example, a star has band flux 1000 units and a companion 1 unit. The ratio is 1/1000 =0.001. Their angular separation is stipulated as 0.2 arcseconds. The ratio and separation answer different questions; neither number specifies a complete image or detection probability.

Suppose an instrument description also gives a nominal 0.1 arcsecond angular scale and an isolated-source limit of 0.5 flux units under stated conditions. These statements alone do not establish that the 1-unit companion will be detected beside the 1000-unit star. The bright source’s extended response, residuals, calibration and noise at that separation matter.

A coronagraph can suppress some bright-source light to help examine surrounding faint structure. Its response and remaining residuals depend on wavelength, separation, instrument and observing conditions. Suppression does not amount to removing every bright-source limitation or guaranteeing every faint companion.

A noise improvement does not automatically sharpen the image

Consider a separate synthetic source and background model with independent Poisson electron totals S and B, exactly known subtracted background mean and other noise terms neglected. The signal-to-noise ratio is S/√(S + B). This is a conditional comparison, not a universal detection rule.

For S =100 and B =100, the ratio is about 7.07. Keeping the same source total but changing B to 900 reduces it to about 3.16. The source need not be less luminous for the measurement to become less precise; its background environment changed.

Under constant source and background rates, four times the exposure changes the latter case to S =400 and B =3600. Its ratio becomes about 6.32, twice the previous value in this model. The longer exposure has not changed the entrance aperture’s diffraction pattern or the detector’s angular pixel interval.

Real observations can add read noise, calibration error, variability, saturation and uncertainty in background estimation. The next lessons examine diffraction, sampling, atmospheric blur and background as distinct parts of the evidence. Improving one part can help a task while leaving another limitation in place.

Worked examples

Try the reasoning

1. Must an unresolved source have an unmeasurable total flux?

   Show the answer and reasoning

   No.

   A finite point response can support a calibrated flux estimate even when source structure remains unresolved.

2. In this lesson, what does contrast 0.001 mean?

   Show the answer and reasoning

   Companion band flux is one thousandth of star band flux.

   The stated convention is companion/star in the same band; other conventions must be identified explicitly.

3. Does four times the exposure automatically halve the aperture’s diffraction scale?

   Show the answer and reasoning

   No.

   Exposure affects the signal and uncertainty model. The optical diffraction scale depends on the aperture and wavelength.

4. Does an isolated-source limiting flux alone guarantee the same faint source can be distinguished beside a much brighter neighbour?

   Show the answer and reasoning

   No.

   The bright source’s response and residual uncertainty at the relevant separation must be included.

What to carry forward

• Angular detail, faint-source measurement and faint-neighbour contrast need different descriptions.
• A source can be unresolved while still providing useful flux and position information.
• State the ratio convention, angular separation, response and uncertainty before interpreting an observing capability.