Starshade Occulter Sizing Calculator

Introduction

A starshade is a large, carefully shaped screen that flies far in front of a space telescope so the telescope can look for faint planets next to bright stars. The basic idea is simple even though the engineering is not: if the starshade blocks the starlight before it reaches the telescope, the telescope can search the darkened region for planets, dust disks, or other dim structures that would otherwise be lost in glare. This calculator gives a first-pass estimate of the starshade diameter needed for a chosen observing wavelength, a chosen distance between the telescope and the starshade, and a target inner working angle, often shortened to IWA. In plain language, the inner working angle is the smallest apparent angle from the star at which the system can begin to see a nearby planet.

The tool is intentionally simplified. Real mission studies include petal shape optimization, broadband diffraction control, alignment tolerances, deployment errors, and fuel limits for formation flying. Even so, a compact sizing relation is still useful because it helps you understand the scale of the hardware very quickly. If you change the wavelength, the required diameter changes. If you move the starshade farther from the telescope, the required diameter changes. If you demand a smaller inner working angle so that planets closer to the star become visible, the starshade usually has to become larger. Those tradeoffs are exactly what this page is designed to make clear.

This calculator is most useful for early concept exploration, classroom demonstrations, and rough mission comparisons. It is not a final optical design tool, but it does capture the main geometric scaling that drives starshade size. That makes it a practical way to build intuition before moving on to more detailed diffraction simulations.

How to Use

Enter the three mission inputs in the form below. The observation wavelength is entered in nanometers, which is convenient for visible and near-infrared astronomy. A value around 500 to 550 nm represents green visible light, while larger values such as 800 to 1000 nm move into the red and near-infrared. The starshade-telescope separation is entered in kilometers because these systems are usually spaced by tens of thousands of kilometers. The desired inner working angle is entered in milliarcseconds, a very small angular unit commonly used in astronomy.

After you click Compute Diameter, the calculator converts all values into SI units, applies the sizing equation, and reports the required starshade diameter in meters. The summary box gives the main result in a compact form, and the table below the explanation is updated with the same values so you can compare runs more easily. If you want to save the result, use the copy button that appears after a successful calculation.

A good way to learn from the tool is to vary one input at a time. Keep the separation and IWA fixed, then increase the wavelength to see how longer-wavelength observations demand a larger starshade. Next, keep wavelength and IWA fixed while changing the separation to see how formation geometry affects the design. Finally, reduce the IWA to represent a more ambitious science goal and notice how quickly the required diameter grows. This one-variable-at-a-time approach makes the trade space much easier to understand.

Formula

The calculator uses a simplified scaling relation that connects starshade diameter, wavelength, separation, and inner working angle. The page originally included MathML, and it is preserved here because it is both machine-readable and accessible in supporting browsers. The relation is presented as an approximation for first-order sizing rather than a complete diffraction model.

The inner working angle in radians is approximated by the wavelength divided by the effective geometric scale set by starshade diameter and separation. In the notation used on this page, the relation is written as:

where is the wavelength, is the starshade diameter, and is the separation between the starshade and the telescope. Rearranging gives the diameter estimate used by the script:

To make the inputs practical, the calculator converts wavelength from nanometers to meters and converts inner working angle from milliarcseconds to radians. The angular conversion is based on:

Once those unit conversions are complete, the script computes the diameter directly in meters. The result should be interpreted as a quick sizing estimate. It tells you the approximate scale of the starshade needed to support the requested observing geometry, not the final petal design, edge profile, or suppression performance across a broad spectral band.

Interpreting the Inputs and Result

Each input has a clear physical meaning. Wavelength represents the color of light you want to observe. Longer wavelengths diffract more strongly, so they generally require a larger starshade for the same angular performance. Separation is the distance between the telescope and the starshade. A larger separation changes the geometry and can reduce the diameter needed for a given target angle, but it also makes formation flying and retargeting more demanding. The inner working angle is the science requirement: smaller values mean the mission can look closer to the star, which is especially important when trying to image planets in compact habitable zones.

The output diameter is best read as a design-scale number. If the calculator returns a value near 20 meters, that suggests a very large deployable structure but one that is still within the range often discussed in mission studies. If the result climbs toward 40 or 50 meters, the concept may still be possible, but deployment complexity, mass, and mission cost become more challenging. If the result is unexpectedly small or large, check the units first. Confusion between arcseconds and milliarcseconds, or between kilometers and meters, can change the answer by orders of magnitude.

Example

Suppose you want to observe at a wavelength of 550 nm, place the starshade 55,000 km from the telescope, and achieve an inner working angle of 60 milliarcseconds. Those values are representative of a serious direct-imaging concept rather than a toy problem. Entering them into the calculator produces a diameter of roughly 37 meters. That number is large, but it is in the same general range as starshade concepts studied for future flagship missions.

The example is useful because it shows how sensitive the design is to science ambition. If you keep the same wavelength and separation but ask for a smaller inner working angle, the diameter rises quickly. If you instead relax the angular requirement, the diameter falls. Likewise, if you move from visible light toward the near-infrared, the required diameter increases because diffraction becomes harder to suppress at longer wavelengths. This is why mission architects spend so much time balancing science goals against deployment limits, propulsion budgets, and manufacturing tolerances.

You can also use the example as a reality check. A result in the tens of meters is normal for ambitious exoplanet imaging concepts. A result in the range of a few centimeters would almost certainly indicate unrealistic inputs, while a result in the hundreds of meters would suggest a mission concept that is far beyond current practical deployment approaches.

Limitations and Assumptions

This calculator is intentionally simplified, so it should not be used as a substitute for a full optical design study. Real starshades are not plain circular disks. They use carefully optimized petal shapes to control diffraction over a range of wavelengths, and their performance depends on edge accuracy, petal count, manufacturing quality, and alignment with the telescope. The simple diameter relation used here does not model those details.

The page also preserves an additional MathML expression from the original content that refers to diffraction suppression scaling under certain assumptions: for petals. This expression is included for continuity with the original educational discussion, but the calculator does not compute suppression level from it. It only estimates diameter from the three user inputs.

Another limitation is that the result is monochromatic in spirit even though the input wavelength can be changed freely. Real observations often span a band of wavelengths, and a starshade sized for one wavelength may not deliver the same performance at much longer wavelengths. Engineers often add margin to the diameter estimate to account for broadband performance, alignment uncertainty, and practical deployment constraints. In early studies, adding a modest design margin after using a calculator like this is common.

Finally, the tool does not estimate fuel use, slew time between targets, telescope aperture effects, throughput, or achievable contrast. Those factors matter enormously in mission planning. What this page does provide is a transparent first-order answer to a foundational question: if you want a certain observing wavelength, a certain separation, and a certain inner working angle, about how large does the starshade need to be?

Why This Trade Study Matters

Direct imaging of exoplanets is one of the hardest tasks in observational astronomy because stars are overwhelmingly brighter than the planets orbiting them. A starshade offers one path around that problem by blocking starlight before it enters the telescope, rather than trying to remove it entirely inside the telescope with internal optics. That difference is why starshades remain attractive in mission studies even though they require precision formation flying over enormous distances.

The numbers produced by this calculator help connect abstract mission goals to physical hardware. If a science team says it wants to detect Earth-like planets close to nearby Sun-like stars, that requirement translates into a small inner working angle. A small inner working angle then pushes the design toward a larger starshade, a larger separation, or both. Once those values are visible, the engineering implications become easier to discuss. Launch packaging, deployment reliability, target-to-target travel time, and mission lifetime all start to come into focus.

That is why even a simplified calculator has value. It turns a difficult optical concept into a set of understandable tradeoffs. Students can use it to learn how diffraction and angular resolution interact. Researchers can use it for quick back-of-the-envelope checks. Enthusiasts can use it to appreciate why future exoplanet missions are so ambitious. In every case, the result is the same: a clearer sense of the scale and challenge of building a giant spaceborne occulter.