From the vantage point of a viewer from Mars . . .
How much would the brightness of the earth vary, based on varying albedo (sun reflecting off water vs. land; existence of snow on land; seasonal axis tilting, etc.)?
Great article.
This question turns out to be a plot point in an Asimov mystery (“The Missing Item”)
It would vary considerably more due to phase and distance. Those tend to work against each other, though: the closer Earth is to Mars, the more of it is “limb” (facing away from the sun and therefore dark). A similar thing happens with Venus from Earth. But the ratio of closest/farthest is much higher for Earth from Mars, so the total brightness as a function of distance would vary more.
It’d be interesting to see the results of calculating this. My guess is that albedo variation would be small potatoes in comparison.
Further reading seems to imply that the greatest brightness is at maximum elongation, which is when the Earth (or any inferior planet) is at half phase, and appears farthest from the sun (by angle, in the sky).
Actually, maximum brightness occurs when the inferior planet is closer than maximum elongation and is showing a fairly distinct crescent aspect. You can see a table of brightness for Venus here.
I believe you are correct. The various surfaces on earth (in ascending order, ocean, forests, desert, cloud, and snow/ice) have hugely different albedos, but from Mars at any given time you would be looking at an entire hemisphere (or at a crescent running from pole to pole), which tends to even things out.
Astronomers track variation in the earth’s albedo from both satellites and from earthshine. Here is a paper (pdf) on the latter. The chart on Page 13-3 shows the albedo over a single night. The albedo varies from 0.26 to 0.32, as the earth rotates and different parts come into view. There is also seasonal variation based on both average cloud cover and ice.
I’m not sure how much the seasonal variation might enhance the extremes. Just using the figures given, we have a 25% increase in reflectivity (and therefore in potential brightness from Mars) from least to most. That sounds like a lot. But it’s only about a quarter of a stellar magnitude, and I can tell you that a difference of a quarter magnitude barely registers with the human eye. As you note, it would be overwhelmed by the more important differences due to distance and phase.
This may be true for Venus as viewed from Earth, but it depends on the relative distances of the two planets, so it won’t necessarily be true for Earth as viewed from Mars. For the limit of an outer planet much further away than the inner planet, the brightest phase is going to be the full one, when it’s almost on the exact opposite side of the Sun, because the increase in distance is going to be insignificant. For the limit of an outer planet only slightly further away, the brightest phase is going to be a very thin crescent, when that crescent is almost right next to you.