## Pi is a special number which doesn't have an exact decimal representation.

Today is Pi Day. Pi is defined as the ratio of a circle's circumference to its diameter, and its mathematical symbol is the Greek letter pi (p). The value of p to the nearest hundredth is 3.14 and today is 3/14 — hence Pi Day.

Pi is a special number which doesn't have an exact decimal representation. The calc program in windows confidently gives p as 3.141592 6535897932384626433832 795, but this is only an approximation. Instead the decimal expansion continues to infinity with a seemingly random sequence of digits.

In one year it will be 3/14/15, reproducing the first five digits of pi, 3.1415. The remaining digits continue, 92653, which could be construed as 9:26:53.

If we ignore the year, we can still have an exact pi time when all of the digits line up. Just use the three digits 159 as the time 1:56 and add all the remaining digits seconds with a fraction part 26.53 5897932384626433832795 ... and you'll have it all. When you consider relativity and quantum effects, it probably doesn't mean anything in a physical sense to specify time too precisely.

The orbits of satellites are computed using pi and some pretty sophisticated mathematics. We can make use of this mathematics without even knowing what is happening. Websites like heavens-above.com, and www.satflare.com/track. asp?q=19549#TOP show us what is happening on a world map.

Bright satellites tonight

The brightest satellite visible tonight is CZ-3 R/B, a Chang Zheng 3B rocket body launched in 2006 into a highly elliptical orbit: 206 km perigee, 15,536 km apogee with an inclination of 28.5 degrees to the equator. It is supposed to get to magnitude 0.9, which is just brighter than first magnitude. It will rise out of the west at 8:31 p.m. and swing low to the south, passing below Orion and Canis Major.

The inclination of 28.5 degrees means that the satellite doesn't appear north of latitude, 28.5 degrees, if viewed from the earth's center. Our own latitude, 28.366284 degrees is probably just within this band, so it might on rare occasions actually be north of us, but not tonight.

Thanks to Kepler's Second Law, we know that the satellite moves faster when it is closer to the center of the Earth (perigee), and slower when it is farther (apogee). In this case, the perigee is in the north, so we know the satellite spends most of its time in the far south at apogee.

Because the Earth is rotating in the same direction as the satellite, its ground track is pushed westward by the eastward motion of the earth's surface. When the satellite slows near apogee, this has a stronger effect and changes the shape of the ground track. The track in the northern hemisphere (closer to perigee) does not match the track in the southern hemisphere (closer to apogee).

This effect is even stronger with tonight's second brightest satellite. The IUS R/B(1) is a rocket body with a more extremely elliptical orbit. It has a perigee of 258 km with an apogee of 24,540 km at 27.0 degrees inclination. Although it doesn't reach the height of a geosynchronous satellite (42,164 km), it is still sufficiently far away that the ground track reverses direction and makes an apparent loop over the Earth.

Tonight's pass is closer to perigee, and hence it will be a bright 1.4 magnitude. It rises at 8:41 p.m. in the west and heads right toward Orion. It passes over Orion's chest at 8:49 p.m.

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