Measuring the distance to Mars with parallax.

[E92M3]

Homework Statement

a) Two observers are separated by a baseline equal to Earth's diameter. If the differences in their measurements of Mars' position at opposition (i think this means when it is closest to earth) is 33.6" (arcsec). What is the distance between Earth and Mars at opposition?

b)If the distance to Mars is to be measured within 10%, how closely must the clocks used by the two observers be synchronized? (Ignore the rotation of the earth. THe orbital velocities of Earth and Mars are 29.79 km/s and 24.13km/s respectively.)

Homework Equations

$$tan\frac{\theta}{2}=\frac{R_{Earth}}{d}$$

The Attempt at a Solution

a)I think I know this part, pls check.
$$tan\frac{\theta}{2}=\frac{R_{Earth}}{d}$$
since theta is small:
$$\frac{\theta}{2}\approx\frac{R_{Earth}}{d}$$
$$d=\frac{2 R_{Earth}}{\theta}$$
Converting theta back into radians I get 7.82x10^10 m.

b) This I'm having trouble with. Well I want 10% error; this means that my error should be 7.82x10^9 m. Observers on Earth will see that Mars is moving at 5.66 km/s (the difference in the orbital velocity between Earth and Mars). And this is how far I got.

 

[anathema]

Hello,

Thank you for your post. I would like to clarify a few things before answering your questions.

Firstly, "opposition" in astronomy refers to the position of a planet when it is on the opposite side of the Earth from the Sun. This is when the planet is at its closest point to Earth in its orbit. So the distance between Earth and Mars at opposition is actually the minimum distance between the two planets, not the maximum.

Secondly, when measuring distances in astronomy, we usually use astronomical units (AU) instead of meters. One AU is equal to the average distance between the Earth and the Sun, which is about 149.6 million kilometers. So, in your first question, the distance between Earth and Mars at opposition would be approximately 0.524 AU (7.82x10^10 m converted to AU).

Now, to answer your second question, we need to use the concept of parallax. Parallax is the apparent shift in position of an object when viewed from different perspectives. In this case, the two observers on Earth are viewing Mars from different perspectives due to the Earth's rotation. This causes a difference in the measured position of Mars.

To calculate the parallax, we can use the following formula:

Parallax = angular separation between the two observers x distance to the object

In this case, the angular separation is given as 33.6" (arcsec), and the distance to Mars is 0.524 AU. So, the parallax would be 0.00462 AU.

Now, we know that the difference in the orbital velocities of Earth and Mars is 5.66 km/s. This means that the time difference between the two observers' measurements would be:

Time difference = Parallax / Orbital velocity difference

= 0.00462 AU / 5.66 km/s

= 8.16 minutes

So, the clocks used by the two observers need to be synchronized within 8.16 minutes in order to measure the distance to Mars within 10% accuracy.

I hope this clarifies things for you. Let me know if you have any further questions.