The Surface Brightness of Nebulae

by Jay Ryan

Okay friends, though the following explanation is a bit long-winded, it should be quite interesting and hopefully educational. As someone who struggled through math class, I've tried to move slowly through the subject, so that anyone not experienced with the subject matter would be able to keep the thread. Enjoy...
    It should be appreciated that galaxies, nebulae and other so-called "deep sky objects" have an intrinsically low surface brightness. Seen with the naked eye or through a modest amateur telescope, these objects appear faint and gray, though some observers insist that slight hints of washed-out color can be seen. This is because the visible light of the objects is not bright enough to activate the observer's color vision, but is only perceived on the observer's "night vision." The vivid colors seen in astro-photographs are the result of an accumulation of light on a photosensitive film, taken over a long period of time.
    As far as I understand the term as used in astronomy, "surface brightness" means "the brightness of the surface" of a celestial object. This term suggests a measure of the quantity of *light flux* emitted from the object, but distributed over the entire apparent *angular area* of the object. So you could think of "surface brightness" as the total amount of emitted light divided by the apparent angular area of the object.

SO WHAT THE HECK IS ANGULAR AREA?

To offer an intuitive example, let's suppose one is viewing from a distance an object that is perfectly square. Imagine a line emanating from the observer's eye to the top of the square. Suppose a similar line emanates from the eye to the bottom of the square so that a triangle is formed with the apex of the triangle at the observer's eye. The angle of the apex, i.e. the angle spanning the two lines, is the angular "height" of the square from top to bottom. Making a similar angle from the left to the right sides of the square yields the angular "width" of the square.
    As is true in regular everyday measurement with a ruler, the "angular area" of this square is the height multiplied by the width. If these angles were measured to be one degree, the square would have an angular area of 1*1=1 square degree. If the square was two degrees on a side, the angular area would be 2*2=4 square degrees, and so on.
    As a practical example of "angular area," the Sun and full Moon are circular objects that (from Earth) each appear to be 1/2 degree in angular diameter. So the "angular area" of the Sun and Moon would be about 0.2 square degrees (according to the formula for a circle, "pi" times "r" squared, where "r" is the radius of the circle, 1/2 the diameter). For elliptical objects like galaxies, the standard formulae for calculating the area of an ellipse would be applied. Other more involved methods would be applied for irregular objects.
    The angular size is a way of determining the apparent size of an object seen from a distance, not its actual "ruler-measured size." This is a technique used to quantify the commonly observed fact that nearer objects appear larger and more distant objects appear smaller. However, if you know the size of the object, you can use simple trigonometry to derive its distance by measuring its angular size. And if you know the distance, you can derive the size the same way. This is what surveyors do on the land. Similar techniques are used to measure objects in the sky, and the sizes and distances of the Sun, Moon, planets, nebulae and galaxies are derived in this way.

THAT WAS BORING AS ALL GET OUT, DOES THIS ACTUALLY HAVE SOMETHING TO DO WITH LIGHT FLUX AND SURFACE BRIGHTNESS?

Back in my physics degree program, we learned about the famous "inverse-square law." This rule derives from Newton's Law of Universal Gravitation of F=G*m(1)*m(2)/r^2. In plain English, this says "there exists a force of attraction between bodies in space proportional to the products of their masses and inversely proportional to the square of the distance of their separation." The inverse-square law is also found to apply *inter alia* to the strengths of electric and magnetic fields. The inverse-square law also applies to light flux. Suppose you see a flashbulb go off at a certain brightness at a certain distance from the camera. If you stand at twice that distance, you will observe the flashbulb to have a light flux of only 1/4 of what you measured at the first distance, since the inverse square of 2 is 1/(2)^2 = 1/4. At four times the original distance, the light flux would be 1/16 and so on. To visualize this natural law, it might be helpful to imagine the flash from the camera as an expanding sphere of light. (This is the common depiction shown in textbooks.) At distances nearer to the camera, the sphere of light would be small (since the radius of the sphere would be the small distance between you and the camera) and at greater distances the sphere would be larger.
    If you are close to the flash, your eye would therefore intercept a larger section of the surface area of the sphere than would happen if you were further away, and you would thus perceive more light flux (and therefore greater brightness). The area of a sphere is proportional to the square of the radius, thus the light flux (or amount of light per unit area) is proportional to the inverse of the square of the radius. So anyway, the inverse-square law simply quantifies the commonly observed fact that, the closer you are to a light source, the brighter it appears. However, the inverse-square law presumes the ideal case of a "point source," i.e. a light source that has no apparent angular dimension, so that all the light flux appears to radiate from an infinitesimal geometric singularity (some "Trek" lingo for you!)
    An ideal point source does not actually occur in nature, but the closest we can come to it is in observing stars. Most stars are so bright and so far away that no disc can be discerned from our terrestrial vantage point. So they can be treated as point sources for most practical purposes, such as star testing a telescope. And since it does follow that stars would get brighter and brighter in magnitude as one would approach them, "surface brightness" would not become an issue until coming sufficiently close that a stellar disc could be resolved.
    Well, no other light-emitting or reflecting objects in nature are point sources, and that includes the nebulae at issue. And it is here that our intuition about brightness perception departs from reality. Magnitude measurements are given for nebulae, but these presume to integrate the entire light emission from the nebula to treat it as through it were a point source with no angular size. Two nebulae might be given as fifth magnitude, but one may appear to have twice the angular area as the other. Thus, the larger nebula will appear much more faint, i.e. having a much lower surface brightness, since the same amount of light is spread out over a larger area of emission.

SO WHY AM I READING ALL THIS CRAP? BRING IT HOME, CHUMP!

Now let's just suppose, for the sake of illustrating my point, that there was a nebula in space at 1000 light years distance that was naked-eye visible from the Earth. Let's suppose, for the sake of simplicity, that this nebula was perfectly square and appeared 1 degree on a side, so that it's angular area was one square degree. Suppose this nebula looked gray and fuzzy to the naked eye and through telescopes, but time-exposure photos showed it to appear in beautiful shades of red, blue and green. Now suppose Rick Sternbach was able to whip up a warp-capable starship that would allow an observer to approach this nebula, in order to view it "up close." It follows from the inverse-square law that this nebula would grow brighter and brighter as the observer drew closer. Intuition would then suggest that the surface brightness of the nebula would also increase to the level where the observer's color perception would be activated, and the bright, vivid neon colors of the nebula would be visible to our observer with the naked eye.
    So our observer friend gets into the spaceship and flies toward the nebula. The observer stops at a distance of 500 light years to take a look, half the distance to the nebula from the Earth. As would have been expected from the inverse-square law, the nebula now appears four times as bright (since 1/r^2 = 1/(1/2)^2 = 4). But here's the rub -- the observer also notices that the angular width is double what it was on Earth. It now appears to be 2 degrees. Same for the angular height. Since the nebula is a square, the angular area of the nebula is now 2*2 = 4 square degrees. So the nebula is four times as bright but it's also four times as large! Since the area increases in proportion to the brightness, the overall surface brightness is the same as was seen from Earth!
    But our observer slept through trig class and had high expectations from a steady diet of space art and Hubble photos and thus didn't understand why the nebula still looked gray and fuzzy. So the ship flies to 250 light years from the nebula, only 1/4 of the distance to the Earth. The nebula is looking really bright since it is 16 times brighter than would be seen from the Earth -- but it's four times as large on each side, or 16 times bigger in area! Again, same surface brightness, none of the expected "Hubble colors."
    So our friend keeps going to only 125 light years away, 1/8 the distance between the nebula and Earth. Now, the nebula is 64 times the brightness as seen from Earth. Since the ship left space dock right after breakfast, our observer finally takes the time to read the morning paper by the light of the nebula, since it now shines so brightly. No doubt this nebula has for thousands of years haunted the mythology of the skies of the nearby worlds! But since it is now has 64 times the angular area, the dang thing is plenty bright, but the surface brightness hasn't changed a whit, still gray and fuzzy!
    After consulting the ship's computer and modeling the phenomenon on the Holodeck using data from Astrometrics, the observer finally realizes that this process will be extrapolated at even closer distances, and no increase in surface brightness will be observed, no matter how close the approach.
    The Holodeck model also indicates that, if the observer flew directly up to the edge of the nebula, it would look dimmer yet, since a "horizon" would result where only a portion of the nebula's surface (and its emitted light) could be seen, but distributed over a nearly 180 degree forward field of view!   The model also shows that from inside the nebula, the situation would be still worse! The observer's ship would be surrounded by the nebula's light, distributed 360 degrees around in every direction. Thus, the full light-emitting thickness of the nebula would no longer be observed as from the outside looking in. The density of the light-emitting nebular gas would thus be "thinned out" in every line of sight direction, thereby reducing the apparent light flux and thus the perceived surface density. Besides, every Starfleet officer knows that sensors are non-operational in a nebula! So our observer heads for home, really bummed out for wasting the whole day on an 875 light year wild goose chase!

WELL, I STILL DON'T LIKE IT! DID YOU CHECK YOUR MATH? HMMMM?