See National Math Standards for this Challenge.

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Q: What was the distance from the Deep Impact spacecraft to the moon on January 16, 2005?

On January 16, 2005, the Deep Impact spacecraft turned its cameras back towards Earth and captured some beautiful images of our Moon. To aid in calibrating the spacecraft instruments, astronomers would like to know precisely how far Deep Impact was from the moon when this picture was taken.

To understand this problem better, we need to know how astronomers locate objects in the sky. Much like navigators who rely on latitude and longitude to pinpoint their location on Earth, astronomers use a pair of numbers to uniquely identify every point on the celestial sphere. These numbers are called Right Ascension and Declination.

Declination is very similar to latitude measurements on Earth. We start by defining a celestial equator, which is simply an extension of the Earth's equator into the sky. This marks zero degrees declination. The range of declination values is from -90° to +90° with +90° situated at the north celestial pole, and -90° situated at the south celestial pole. Each degree is divided into 60 arcminutes, and each arcminute is divided into 60 arcseconds.

Right ascension is measured along the east-west direction, much like longitude on Earth, except it is measured in "hours", rather than degrees. The range for right ascension (or R.A.) is from 0h to 23h, incrementing as one looks east across the sky. Each hour is divided into 60 minutes, and each minute is further divided into 60 seconds.

As an example of how this is used, let's return to the beginning of the problem. At the time the moon picture was taken, Deep Impact was located at 13 hours, 12 minutes, 31.27 seconds right ascension (also written as 13h 12m 31.27s R.A.) and -3 degrees, 26 arcminutes, 38.2 arcseconds declination (also written as -3° 26' 38.2" Dec.). We also know that the spacecraft was 0.0084401211 AU from Earth (1 au = 1.5 x 10⁸ km). An astronomical unit (AU) is a unit of distance that is approximately equal to the mean distance between the Earth and the Sun.

We can look up the same information for the moon at a particular time. The data can be retrieved via the JPL Horizon's webpage located at the following URL: ssd.jpl.nasa.gov/?horizons

The celestial coordinates for both Deep Impact and the moon are summarized in this table:

These data are sometimes referred to as the "ephemeris" of the objects. Now it's your turn to solve a problem of real interest to astronomers. Given the ephemeris data for Deep Impact spacecraft and the moon, plus some geometry and what you've just learned about Right Ascension and Declination, can you figure out how far the spacecraft was from the moon on January 16th?

Setting up the problem:

We can describe the location of an object in space (call it P) with the following vector notation:

Vector from σ to point P (representing any point in space) is defined by:
r = magnitude
φ = angle with respect to xy-plane
θ = angle with respect to positive x-axis

These coordinates, with the Earth located at the origin (σ) relate to the ephemeris data as follows:
r = the distance from the Earth to the object we're interested in
φ = declination of the object
θ = right ascension of the object

In this problem, we are interested in finding the distance between two points in space, which we can call P₁ and P₂. These will represent the location of the Deep Impact spacecraft and the moon, respectively.

To find the distance between any two points in space, we can use the distance formula:

where (x₁, y₁, z₁) and (x₂, y₂, z₂) are the Cartesian coordinates of the two points.

To use this, we need to first convert our (r, φ, θ) coordinates to (x, y, z) coordinates.

rz = projection of r onto z-axis = r sin φ rxy = projection of r onto xy-plane = r cos φ

rx = projection of rxy onto x-axis = rxy cos θ ry = projection of rxy onto y-axis = rxy sin θ

r_x, r_y, r_z are going to be the (x, y, z) coordinates we need.

r_xy = r cos φ
r_x = r_xy cos θ = (r cos φ) cos θ
r_y = r_xy sin θ = (r cos φ) sin θ

x = r cos φ cos θ y = r cos φ sin θ z = r sin φ

These three equations will give us the (x, y, z) coordinates if we have the distance, right ascension, and declination!

Before we use these transformations, we need to convert right ascension and declination to degrees (or radians).

R.A. recorded in hours, minutes, seconds
Dec. recorded in degrees, arcminutes, arcseconds

For right ascension: 360° = 24h = 1440m = 86,400s or,

1h = 360°/24h = 15°

1m = 360°/1440m = (1/4)°

1s = 360°/86400s = (1/240)°

For declination: 1° = 60' = 3600" or,

1' = (1/60)°

1" = (1/3600)°

We now have all the conversion and transformations we need to solve the problem!

Why did the science team need to make this calculation?

In addition to calibrating the instruments, we wanted to give the image of the Moon some perspective for the viewer. We were writing figure captions for release to the public. Ordinarily, the distance is calculated by the data processing pipeline, a series of computer programs that operates on the returned data as well as information from the spacecraft about its condition and location in space. At the time when the caption was being written, we did not have all of the information we needed in the data pipeline. It was still being set up. So we had to make the calculations ourselves. We were glad that we had studied trigonometry and pre-calculus so that we could do the calculation. Then we were also able to check that the computer code, when it was available, had been correctly programmed.

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A: Given facts you will need:

Deep Impact: rd = 0.00844 AU RAd = 13h 12m 31.27s Decd = -3° 26' 38.2"

Moon: rm = 0.00257 AU RAm = 1h 22m 47.56s Decm = 7° 18' 12.5"

Convert R.A., Dec. for Deep Impact:

RA: θ_d = 15 (hours) + 1/4 (minutes) + 1/240 (seconds)

   = 15 (13) + 1/4 (12) + 1/240 (31.27)

θ_d = 198.1°

Dec.: φ_d = (degrees) + 1/60 (arcminutes) + 1/3600 (arcseconds)

   = (-3) + 1/60 (26') + 1/3600 (38.2")

φ_d = -2.56°

Now do the same for the moon:

RA: θ_m = 15 (1) + 1/4 (22) + 1/240 (47.56)

θ_m = 20.70°

Dec.: φ_m = (7) + 1/60 (18') + 1/3600 (12.5")

φ_m = 7.303°

So, now we can write the (r, φ, θ) coordinates for both the moon and the spacecraft:

Deep Impact:

r = 0.00844 AU φ = -2.56° θ = 198.1°

Moon:

r = 0.00257 AU φ = 7.303° θ = 20.70°

Let's use these numbers with our transformation equations to get the x, y, z coordinates:

Deep Impact:

x = r cos φ cos θ = (0.00844) cos(-2.56°) cos(198.1°) = -0.0080

y = r cos φ sin θ = (0.00844) cos(-2.56°) sin(198.1°) = -0.0026

z = r sin φ = (0.00844) sin(-2.56°) = -0.00038

Moon:

x = r cos φ cos θ = (0.00257) cos(7.303°) cos(20.70°) = 0.0024

y = r cos φ sin θ = (0.00257) cos(7.303°) sin(20.70°) = 0.0009

z = r sin φ = (0.00257) sin(7.303°) = 0.00033

One step away from getting the distance!

This distance is in Astronomical Units (AU). We can use the conversion 1 AU = 1.5 x 10⁸ km to convert the distance:

0.0110 AU x (1.5 x 10⁸ km/1 AU) = 1.65 x 10⁶ km

That's how far Deep Impact was from the moon on 1/16/05!!

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