### Overview

The "Pi in the Sky" math challenge gives students a chance to take part in recent discoveries and upcoming celestial events, all while using math and pi just like NASA scientists and engineers. In this problem from the set, students use the mathematical constant pi to calculate how long it will take an Earth satellite to image the entire planet.

### Background

Soil Moisture Active Passive, or SMAP, is a remote sensing observatory that carries two instruments designed to map soil moisture and determine the freeze or thaw state of the same area being mapped. Soil moisture content is mapped via a combination of radar and radiometer measurements, while the freeze/thaw mapping is accomplished using unique properties of the radar system’s measurements. Both the radar and radiometer share a common antenna and feed assembly externally, but their electronics inside SMAP are different. When combined, the SMAP radar and radiometer are designed to deliver high accuracy, high resolution global maps of the Earth’s soil moisture and freeze/thaw state.

The radar sends pulses of radio waves down to a spot on Earth and measures the echo that returns a few microseconds later. The strength and “shape” of the echoes can be interpreted to indicate the moisture level of the soil, even through moderate levels of vegetation. Since the radar actively sends and receives radio waves, this is where the ‘active’ in SMAP comes from.

The radiometer detects radio waves that are emitted by the ground from the same small area. The strength of the emission is an indicator of the temperature of the ground in that location. Since a radiometer passively makes these temperature measurements, this is where the ‘passive’ in SMAP comes from.

The large antenna that is shared by both radar and radiometer is SMAP’s most prominent feature. It operates just like a satellite dish where a reflector collects all of the radio waves and focuses them into the feedhorn. Like a funnel, the feedhorn collects the echoes from the radar and the surface emissions from the radiometer and sends them to the radar and radiometer electronics for processing images. The feedhorn is a large cone visible on the outside of the spacecraft. The antenna reflector is a mesh antenna 6 meters (about 20 feet) in diameter.

In order to achieve large coverage on the ground the antenna spins at 14.6 revolutions per minute (one revolution every 4 seconds). SMAP’s orbital motion combined with the spin of the antenna sweeps a small field of view in a series of overlapping loops that create a swath 1000 km (621 miles) wide.

### Procedures

1. The Soil Moisture Active Passive, or SMAP, satellite is designed to image 1,000-kilometer-wide swaths of Earth from a near-polar, sun-synchronous orbit 685 kilometers above Earth’s surface. How many days will it take SMAP to image all of Earth’s surface? (You may disregard any overlap that may occur.)

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