Question

On August 27,2003 , Mars approached as close to Earth as it will for over 50,000 years. If its angular size (the planet's radius, measured by the angle the radius subtends) on that day was measured by an astronomer to be 24.9 seconds of arc, and its radius is known to be \(6784 \mathrm{~km}\), how close was the approach distance? Be sure to use an appropriate number of significant figures in your answer.

Short answer

**Answer**: The distance between Mars and Earth during the close approach is approximately 56,162 km.

Step-by-step solution

Visualize and understand the problem

The problem describes a right triangle, with Earth as the vertex, the distance from Earth to Mars as the adjacent side, and the radius of Mars as the opposite side. We can relate the angular size of Mars, the distance between Mars and Earth, and the radius of Mars using the tangent function of trigonometry.

Convert the angular size from seconds of arc to degrees

Since the given angular size is in seconds of arc, we need to convert it to degrees to use trigonometry. There are 3600 seconds of arc in 1 degree, so divide the angular size by 3600: $$ \text{angular size (degrees)} = \frac{24.9}{3600} = 0.006916 \mathrm{^\circ} $$

Find the tangent of the angular size

To apply the tangent function, we first find the tangent of the angular size (in degrees): $$ \tan(0.006916 \mathrm{^\circ}) = 0.00012081 $$

Use the tangent function to find the distance between Mars and Earth

Now, we use the tangent function to relate the angular size, the distance between Mars and Earth (adjacent side), and the radius of Mars (opposite side). We can write: $$ \tan(\text{angular size (degrees)}) = \frac{\text{radius of Mars}}{\text{distance between Mars and Earth}} $$ Rearrange this to solve for the distance between Mars and Earth, given the tangent of the angular size and the radius of Mars: $$ \text{distance between Mars and Earth} = \frac{\text{radius of Mars}}{\tan(\text{angular size (degrees)})} $$ Substitute the known values: $$ \text{distance between Mars and Earth} = \frac{6784 \mathrm{~km}}{0.00012081} \approx 56,162,100 \mathrm{~km} $$ Therefore, the distance between Mars and Earth during the close approach was about 56,162,100 km, which should be reported with the appropriate number of significant figures, 5 in this case, as 56,162 km.

Key concepts

Angular Size

In astronomy, angular size is a critical concept as it helps scientists and observers determine the size of an object in the sky from a particular observing point.
Angular size describes the span of an object's diameter, angle-wise, when viewed from a certain point. Imagine holding out your thumb at arm's length; the width of your thumb would cover a certain part of your field of view. This 'coverage' is akin to the angular size of your thumb from that distance.

When it comes to celestial bodies, such as planets or stars, they appear to us as points without any discernible size due to their vast distances from Earth. Measuring their angular size, however, can give us a relative measure of their actual size or distance. Astronomers use units like degrees, arcminutes, and arcseconds, with one degree equalling 60 arcminutes, and one arcminute equaling 60 arcseconds.

When the angular size is known, and we have the actual size of the object, we can employ trigonometry to estimate the distance to the object. This is precisely what is done to calculate the distance of planets, like Mars, from Earth.

Trigonometric Functions

Trigonometric functions are the foundation of trigonometry, a branch of mathematics that examines the relationships between the angles and sides of triangles. These functions include sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).

In the context of calculating astronomical distances, the tangent function is often used. It is defined, in a right-angled triangle, as the ratio of the length of the opposite side to the length of the adjacent side. This function becomes invaluable when the angular size of a celestial object (like Mars in our exercise) is known, as it allows us to establish a link between the angle subtended by the radius of the object and the actual radius of the object to find the distance.

The viable formula we use is of the form \(\tan(\theta) = \frac{O}{A}\), where \(\theta\) is the angle, O is the length of the opposite side to the angle (e.g. the radius of Mars), and A is the length of the adjacent side (the distance from the observer to Mars). Through manipulation of this equation, we can solve for unknown distances in astronomical calculations.

Significant Figures

Significant figures are a way of expressing precision in numerical answers. These are the meaningful digits in a measure or calculation that contribute to its precision. The rules of significant figures dictate how many digits should be retained in the results from calculations.

When dealing with astronomical calculations, the principle of significant figures becomes essential to accurately report distances. It's important to start with appropriate precision in measurements and maintain this throughout the calculation process. In reporting our final answer, we round to the least precise measurement we've used.

For instance, in the given exercise, the measured angular size of Mars was reported with three significant figures (24.9 seconds of arc), which influences the precision of our final answer. This is why, although our calculations yielded a long decimal number for the distance, we rounded it to five significant figures, aligning with the precision of our measurements and calculations. Appropriately using significant figures ensures that the reported distance of 56,162 km does not imply undeserved accuracy.