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How to calculate distance to an asteroid

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Michael Richmond

Nov 3, 2001

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Step 1: Two sites on Earth

First, one must have two observatories on Earth
from which to make simultaneous measurements.
Let's call them
**R**, for Rochester,
and **A**, for Annapolis.
Set up a coordinate system in which the

- x-axis is in the plane of the equator, due South as seen from Rochester
- y-axis is in the plane of the equator, due East as seen from Rochester
- z-axis runs up through the Earth's North Pole (and Celestial Pole)

The vector distance between the two sites is
**V = R - A**,
shown by the green arrow in the figure.
One can calculate **V** if one knows
the latitude and longitude of each of the two sites.
The angle **alpha** is the difference in longitudes.

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Step 2: Observe an asteroid

From each site, observe the same asteroid, simultaneously.
Measure the position of the asteroid relative to field
stars.
Find the shift in the asteroid's apparent position
as seen from the two sites.
Call that angular shift **theta**, the parallax angle.

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Step 3: Determine a vector pointing towards the asteroid

We make a slight approximation in this step -- it results
in a negligible effect in the final results.
The (RA, Dec) position to the asteroid is very nearly
the same from both sites; it typically differs
by only a few arcseconds. In fact, the (RA, Dec) position
would be very nearly the same as seen from ANY place
on Earth.
Therefore, we can approximate the unit vector **Q**,
which runs from Rochester towards the asteroid
as the unit vector running from the center of the Earth
towards the asteroid.
That vector **Q** is composed of two angles:
the asteroid's Declination, and its Hour Angle as seen
from Rochester.
Remember, Rochester defines the x-axis of our coordinate
system, and the Hour Angle is exactly the projection
onto the x-y plane
of the angle between **Q** and the x-axis.

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Step 4: Calculate the effective baseline

We can't use the entire baseline **V** between the two
observatories unless it was perpendicular to the
direction of the asteroid **Q**.
Instead, we need to find **B**, the component of the
baseline **V** which is perpendicular to **Q**:

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Step 5: Use trigonometry to determine the distance to the asteroid

Now that we know the effective baseline **B**
and the parallax angle **theta**,
we can use calculate the distance to the asteroid easily.

The distance to the asteroid is

0.5 * B
dist = -------------------
sin (0.5 * theta)

For the tiny parallax angles we typically measure,
one can incur no significant error by using the simpler formula

B
dist = ------------
sin (theta)