An occultation occurs when a nearby celestial object passes in front of a more distant one and completely hides it from view. In a lunar occultation, the Moon passes in front of a star or planet. Lunar occultations of faint stars occur all the time, but they're hard to see because the Moon's light swamps dim objects. Lunar occultations of stars brighter than 8th magnitude can be easily observed with small telescopes. These events can provide dramatic evidence concerning the angular sizes of stars, and yield information useful in refining our knowledge of the Moon's orbit around the Earth.
This table provides data for lunar occultations visible from Honolulu during Astr110L class meetings in Fall 2005. Predicted times are rounded to the nearest minute. The altitude and azimuth columns give the Moon position at that time. The illumination column gives the percentage of the Moon's disk which is illuminated; the trailing `+' sign indicates the moon is waxing. The magnitude column gives the apparent magnitude of the star; events involving stars of magnitude 8.0 or fainter are difficult to observe.
|Thur.||10/06/05||19:43||9||240||14+||5.0||bright star, low altitude|
|Thur.||11/10/05||20:04||61||168||74+||4.9||time of reappearance; occultation begins at 18:42|
If we observe an occultation from two or more locations separated by at least a few km along an East-West direction, we can estimate the speed of the Moon's shadow as it sweeps across the Earth. This works best if the Moon is fairly high to the South when it occults the star; in other words, the Moon's azimuth should be close to 180°. In the table above, azimuths near this value are shown in boldface; they are best for this purpose, but any observation will be better than none!
For example, consider the occultation on 10/12/05 at 20:24 HT. The occultation will occur about 20 sec earlier at Kapiolani Park than at Sandy Beach. The East-West separation of these two sites is about 15 km, implying the Moon's shadow moves from West to East at 0.75 km/sec relative to the Earth's surface. To this we should add the Earth's rotational velocity, which is 0.44 km/sec at Oahu's latitude, to get the speed of the Moon along its orbit. The result is 1.19 km/sec, as compared to the Moon's mean orbital speed of 1.02 km/sec. Much of the ~16% discrepancy in these speeds can be resolved by noting that the Moon is close to perigee and therefore, by Kepler's second law, moving faster than average in its orbit. A better calculation might also take account of the North-South separation of Kapiolani Park and Sandy Beach and the North-South component of the Moon's motion, but such refinements don't add much to the basic concept and would be hard to implement in this class.
Just how fast is a star's light cut off by the edge of the Moon? If the star were large enough to appear as a disk, and not just a point of light, you'd see it fade out gradually as the Moon covered it up. But in fact, the star will vanish in a split second - you will definitely not notice it fading out gradually.
We can use this fact to make a very rough estimate of the distances to stars. Let's say the star takes less than 0.1 sec to vanish (this is about the shortest time we can easily perceive). Let's also say that the star has the same diameter as our Sun, which is 1.4×106 km. These are the only assumptions used in this estimate; neither is very accurate, but they are OK for a very rough answer. In particular, they will serve to find the smallest distance the star could possibly have.
As seen from Earth, the Moon moves with respect to the stars at an average rate of 0.00015°/sec (360° in 27.3 days); in other words, each second it's position changes by 0.00015°. So, if the star takes less than 0.1 sec to fade out, it must have an angular diameter which is less than one-tenth of this angle, or <0.000015°. In other words, 0.000015° is an upper limit for the star's angular diameter - we don't know the true value, but we do know that it is less than 0.000015°. (This is about 20 times smaller than anything we can see with our telescopes; in fact, even the most powerful telescopes have trouble seeing detail this small!)
Now, since we have assumed how big the star really is, and we have an upper limit for how big it appears to be, we should be able to calculate a lower limit for its distance. The equation required is the same one used for parallax distances:
Here we use our guess for the star's actual diameter (1.4×106 km) for the baseline b, and our upper limit for the star's angular diameter (0.000015°) as the angle . The result for the star's distance D is about 5.3×1012 km, or 0.55 l.y. (light-years); remember that this is a lower limit for the distance, so that the actual distance can be much greater. In fact, the nearest stars are nearly 10 times further away. Still, this is at least a rough figure for the distance to a star; it's pretty good for an estimate made using just a small telescope (or just your naked eyes, if the occulted star is sufficiently bright).
We can express the formula in another way: remember that in class we introduced the relation arc = angle x distance, with the angle in radians. From this relation we get distance = arc/angle. It should be clear that the "arc" is the size of the Sun; you can verify that we get the same result as before.
One last comment: the distance estimate we have obtained, although too small compared to the real distances to the stars, is large enough to argue that we cannot detect stellar parallaxes with our naked eyes; so the failure to detect stellar parallaxes in ancient Greece (or Egypt) was not a strong argument against the idea that the Earth must move around the Sun. Unfortunately, it seems that the Greek astronomers who defended the heliocentric ideas never thought of arguing in terms of the sudden disappearance of stars behind the Moon.
Provides information on upcoming occultations by the Moon, planets, and asteroids.
Make the observations described above, and write a report on your work. This report should include, in order,
In more detail, here are several things you should be sure to do in your lab report:
This report is due in class by the end of November, in principle. More details later.
Last modified: October 5, 2005