The light map: one map of the sky's light that every frame is checked against

A thin line beside a passage says how far its claim is proven: published, prototyped, planned, illustrative. Colored numbers carry the same mark. A passage with no line is stated. How claims are marked.

What the light map is

It is an all-sky brightness map on HEALPix's equal-area grid: the sky cut into cells of equal area, with a layer for each kind of light. One layer holds stars, one galaxies, one the glow of nebulae. SkyCruncher's app calls this the light map. The app is in development, and the map is built in my test bench and in the app's new pipeline, not in anything that has shipped. (prototyped)

The map holds only what the sky holds: where each source is and how bright it is. Nothing about a telescope is stored in it: no focal length, no pixel size, no star width, no exposure time. Those belong to each frame, and a frame brings them when it asks the map a question. (prototyped)

The idea started as a question about speed. Working out how much of a star's light really belongs to its neighbors is a weighted sum over nearby stars, which is the kind of sum a graphics card is built for. Storing the sky's light once, and letting each frame ask the map what lands on it, replaced numbers that had been worked out ahead of time for one assumed star width.

Why the sky is cut into equal-area cells

The obvious grid for the sky is the one on a globe: lines of right ascension and declination, like longitude and latitude. Its cells are not the same size. They shrink toward the poles, where the lines meet, so a cell near the celestial pole covers a sliver of the sky a cell at the equator covers. Any count or brightness per cell then means something different depending on where on the sky you are.

HEALPix (Hierarchical Equal Area isoLatitude Pixelization, Górski and colleagues, 2005) fixes that. It cuts the whole sphere into 12 base cells of exactly equal area, and every cell splits into four children of equal area again, as many times as you like. Maps of the cosmic microwave background from the WMAP and Planck satellites use it, and Gaia, the European Space Agency's star-mapping mission, builds a HEALPix cell number into every star's catalog ID. (published)

Each round of splitting is one order. Because every cell splits into exactly four, the cells nest: in HEALPix's nested numbering, the children of cell n are numbered 4n to 4n + 3. Zooming in or out is arithmetic on the cell number, and the light in a cell is exactly the light in its four children added up. That nesting makes the map a tree, so a question can stop at whatever level of detail it needs.

Computed from the grid's definition: the sky's 41,253 square degrees shared among 12 × 4^order cells; the width is the square root of one cell's area. An arcminute is 1/60 of a degree and an arcsecond 1/60 of an arcminute.

For scale: the full Moon is about 31 arcminutes across, so an order-10 cell is about a ninth of the Moon's width, and an order-16 cell is about the size of one Seestar pixel (2.3 to 4.0 arcsec, by model, from the models' specs).

What the map holds in each cell

Every layer is filed under the same nested cell numbers, so "what is in this patch of sky" is one lookup per layer. (prototyped)

• Stars, one by one. Every star from the Gaia catalog, stored once: its position, finer than any cell, and its brightness in Gaia's three bands (G, and the blue and red halves BP and RP). Gaia's own star numbers already follow its cells, so the catalog arrives nearly in the map's order. (prototyped)
• Starlight per cell. The total starlight in each cell, at every order from whole-sky cells down to a few arcminutes, each the sum of its children. A coarse question, such as how much starlight a wide frame's corner holds, has a cheap answer without touching single stars. (prototyped)
• Galaxies, as shapes. Each galaxy as a few numbers: where it is, its outline as an ellipse, how its light falls off from the center, and its brightness. A frame paints the galaxy from those numbers at its own sharpness, which is far smaller than storing a picture of it. (prototyped)
• Nebula glow. The red hydrogen-alpha glow of nebulae, which is smooth rather than a list of points. The coarse cells hold its mean, and each cell below holds only how its four children differ from it. A difference no bigger than the survey's own noise is dropped, so the map spends detail only where a nebula has structure, the way image formats compress a photograph. (prototyped)
• Stars too faint to list. Past the catalog's limit there are still stars, too faint to list one by one, and together they add a faint glow. A layer for it, estimated cell by cell from how the counts of listed stars run out, is designed but not built. (planned)
• Dust. Dust in our galaxy dims and reddens the light behind it, and also scatters starlight into a faint glow of its own. Both are planned as layers on the same cells. (planned)

What the map leaves out on purpose: the sky's own glow. Light pollution, moonlight, twilight and haze change every night and from place to place, so they belong to each frame, not to the sky.

Every layer has to be on the same grid in the same sky coordinates. Some published all-sky maps come on HEALPix cells in galactic coordinates, which line up with the Milky Way instead of the celestial equator. The cell numbers look the same, but they point at different sky, so such a map is resampled onto the light map's cells once, when the map is built.

How one frame's stars land on the map

A frame brings three things: its pose (where it points, how it is turned, and its scale), its lens model (how its optics bend straight lines on the sky toward the corners), and its star shape (how its focus, the air and its tracking spread a point of light, which can change across the frame). With those, the map can draw what the frame should have recorded, before anything on the frame is measured. (prototyped)

1. Take the cells under the frame, plus a margin, because a bright star just outside the edge still throws light in. (prototyped)
2. Move each star from its sky position to a pixel position through the pose and the lens model. (prototyped)
3. Spread each star's light by the frame's own star shape, and paint the galaxies and nebula glow the same way. The sum is a predicted frame: the sky's light as this telescope, on this night, should have seen it. (prototyped)

The predicted frame then does three jobs. (prototyped)

• Finding the pose. In the app's new pipeline, a frame's pose and lens model are fitted by adjusting them until the predicted frame matches the real pixels, every pixel at once. This is still being tuned on real frames. It works after plate solving has found roughly where the frame points. (prototyped)
• Neighbor light, with names. When you measure a star, the patch you measure also holds some of its neighbors' light. That borrowed light is a sum with one term per neighbor, so each share has an owner: this Gaia star, that galaxy, the nebula behind it. A measurement can carry its biggest owners by name, not only a total. (prototyped)
• The stars the frame didn't detect. The map knows about stars too faint for the frame to pick out, which still add light around brighter ones. Predicting them is how their light gets counted at all. (prototyped)

Exposure time drops out of the neighbor share. Both stars collect light for the same seconds, so the share is a ratio of two brightnesses in one frame, and in expectation the seconds cancel. A longer exposure still matters indirectly: more neighbors are detected and measured, bright ones may saturate, and a small bias stands out more as the noise falls.

How it lets frames from different nights and telescopes be compared

This is the reason for the whole design. My catalog mixes telescopes: Seestar S30, S30 Pro and S50 units, other cameras and lenses, my frames and frames the community shared. Their pixels cover different amounts of sky, their star shapes differ, and so do their nights. Comparing two of their measurements of the same star directly mixes up the star with the telescope.

• One sky, read many ways. The map has no telescope built in, so each frame is compared with the same sky, drawn through its own pose, lens, star shape and filter. What is left over, the real frame minus its prediction, is in that frame's own terms, and leftovers from different telescopes can be set side by side. (prototyped)
• The cells stay put. A cell number means the same patch of sky in every frame, on every night, from every telescope. Frames move across the cells; the cells don't move.
• Each frame reads at its own level of detail. A wide, soft frame reads coarse cells and a sharp one reads fine cells, from the same tree. The Seestar models differ in how much sky a pixel covers, and none of them needs a map of its own. (prototyped)
• Neighbor light follows the frame. The same star borrows more light on a soft night than on a sharp one, and more through a wide-field lens than through a Seestar. Because the share is worked out per frame, at that frame's own star shape, a soft night's measurement can be corrected rather than thrown away. (prototyped)
• Stars move. Gaia's positions are given for the start of 2016, and nearby stars drift measurably across the sky in a few years, so a star's position has to be carried to the date a frame was taken before it is placed.

The same cells also file what comes out of measuring: what each frame recorded is stored against the cells it covered, and its frame receipt records what ran and what it measured. Two people who shot the same field years apart end up filed under the same cells. (planned)

What isn't done yet

• Measured results. This Note explains the design only. How well the map's predictions match real frames, per layer, will go on dated Experiments once the checks are final, each with the frames it used. (planned)
• The faint-star and dust layers are designed but not built. (planned)
• Outside surveys. The galaxy and nebula layers are built from published surveys. I'm still checking each survey's terms, so nothing made from them is published here yet.
• The app hasn't shipped. It is in development, with no release date.

Read more

• Plate solving: how a frame first finds roughly where it points.
• Frame receipt format: what gets recorded about each frame. A receipt is that record: what ran, what it measured, and what it could not.
• What differs between the Seestar models: pixel scale and field for each.
• Starlight, Measured: the exhibit, following light from the star to a number.
• All Learn Notes

Star data: ESA/Gaia/DPAC, Gaia DR3 (CC BY-NC 3.0 IGO). HEALPix: K. M. Górski et al., 2005, The Astrophysical Journal 622, 759.