Close Encounter of Mars and Earth? (by Dr JB Lim)

How big would Mars be when she is closest to the Earth when compared to a Full Moon at her closest?

Summary:
• Mars will not be closest to Earth on this August or at every August
• Even when Mars is theoretically closest to Earth she will still be 54.6 million km apart from each other
• Even at her closest, Mars will just be 25.67 arc sec in apparent diameter
• The Full Moon will still be 77 times larger than the apparent size of Mars then
• The Internet claim circulated round the world is a fraud

Full Explanation:

Now allow me to write my astronomical explanations using simple mathematics.

Do not be taken in by Internet hoaxes that Mars will be seen as big as the Full Moon on every August of each year.
We should never accept every thing that is given in the Internet. Always consult a qualified specialist or a professional in that area of expertise. Not all information given in the Internet is true and reliable. Any Dick, Tom and Harry can write just about anything rubbish inside a website, or forward it to an e-mail address and ask you to circulate it round.
Always examine the credentials of the writer.
Always use scientific logic and mathematical analysis to gather evidence-based conclusion. Use your intelligent human brain.

Now, let me explain below:
The distance from Earth to Mars varies minute to minute as they orbit around the Sun. The two travel along elliptical orbits that are not harmonized, such that they are constantly changing their positions, between getting closer together and going farther apart.
Under the best orbital circumstances, Mars and Earth are closest together when Earth is at her farthest point from the Sun (aphelion) and Mars is at its closest to the Sun (perihelion). At that point in orbit the two worlds would be within 54.6 million km of each other.
Even then, this point in orbit at best is theoretical because it has not been observed in recorded history. The closest known approach in recent years was in 2003, when Earth and Mars were separated by only 56 million km.
On the opposite end of the scale, Mars and Earth can be as far as 401 million km apart when they are in opposition and both are at aphelion. The average distance between the two is 225 million km.
Now let us assume on a theoretical basis that both Mars and Earth can come so close together in an unprecedented planetary history. What then would be the distance separating them, and the apparent disc size of Mars?

The Calculations:
The closest theoretical distance possible can be calculated out from some of the relevant planetary data below would be: (206 669 000 - 152,098,232) = 54 570 768 km (54.6 million) km.
Thus the apparent diameter of Mars as seen from Earth when they are at their theoretically closest distance would be:
tan θ = Diameter of Mars ÷ Distance between Earth and Mars
= 6792.4 km ÷ 54 570 768 = 1.24469 x 10 -4 Hence, θ = tan-1 (1.24469 x 10-4) = 7.13158 x 10 -3 degree
= 0.4278948584 arc min
= 25.67 arc sec Hence apparent angular size when Mars is closest to Earth is only 25.67 arc sec What about the Full Moon then when she is at her closest to Earth?

Again we use the relevant planetary data below to calculate:

Moon apparent angular size at its closest = arc tan ω = 3476.28 / 362 570 = 0.5493 degree x 60 = 32.9597 arc minute = 1977.58295 arc second

This means the Moon is 77 times larger than Mars when both are closest to Earth.

Mars at its closest to Earth is 0.01298 times smaller than the Full Moon when the Moon is at her closest to Earth.

Planetary data:
Mars: Aphelion 249,209,300 km
Perihelion 206,669,000 km
Eccentricity 0.093 315
Equatorial radius 3,396.2 ± 0.1 km
Equatorial diameter: 6792.4 km
Mars closest distance to Earth: 54.6 million km (54 600 000 km)

Earth: Aphelion: 152,098,232 km
Perihelion: 147,098,290 km
Eccentricity: 0.01671123
Equatorial radius: 6,378.1 km
Equatorial diameter: 12756.2 km

Moon: Equatorial radius 1,738.14 km (0.273 Earths)
Equatorial diameter: 3476.28 km
Perigee 362,570 km (0.0024 AU)
(356,400-370,400 km)
Apogee: 405,410 km (0.0027 AU)
(404,000-406,700 km)
Semi-major axis 384,399 km (0.00257 AU)
Eccentricity: 0.0549

I hope I have mathematically proved my point.

My inability: However, I am unable to tell you when Mars or any other planet will come closest to Earth or to each other because all the planets are continuously moving around the Sun and their positional data are changing endlessly. In order to determine their positions with each other, it requires an enormous computation power, and this is well beyond me using manual calculation even on the most advanced scientific calculator. I will not be able to handle such calculations where the parameters are continuously changing due to motions using only my simple calculator. Normally, such computations are handled by huge main frame computers that have very powerful computing speeds. Computers such as main frame and supercomputers that are capable of billions or trillions of calculations per second would be able to simulate the continuous motions of each planet simultaneously relative to the motions of other planets.

Mathematical Iteration: Only a supercomputer is capable of simulating such complex motions. This probably requires a very highly advanced mathematical procedure called ‘iterations’ where the continuously changing variables of motions are put back into the equations as they are calculated out, and then put back again and again into the equations for ‘updating’. The motions of all the planets are continuously moving and their orbital data are all changing at every count of a second. This is not going to be easy task to calculate. This type of mathematical procedure can only be handled by a super fast computer, the faster the computer, the better the results. This is no job for a mere human brain. Mathematical iteration is used in astronomy especially to calculate out the nuclear reactions going on near the speed of light in the interior of stars. A hyper computer is needed to catch up with the speed of light.

Please do not confuse this term ‘iteration’ meaning ‘to repeat’ with the term ‘integration’ meaning ‘to put together’ in calculus as in integral calculus.

No one has a brain that can work at a dazzling speed of a supercomputer. Neither can my simple pocket calculator able to handle such astounding motions. Sorry for this reason, I cannot give you the answer when Mars and Earth will next come closest together.

My ability: However, if you merely ask me where in the sky or in deep space Mars, Earth, or any planet would be at a specific given time and date; I will be able to calculate this for you. That is not a problem. Even though this is a horrendously long and complex calculation it can be done if you have a powerful scientific calculator with all the data of orbital elements ready for input. I should be able to do this within 1-2 days through non-stop calculations on a powerful scientific calculator. But as far as that e-mail claim that Mars will be as big as a Full Moon on August is concerned, that is a hoax of astronomical dimension that has been circulating round the world for the last 5 years since 2006.

I feel sorry for that hoax writer who has his fraud circulated to me - the wrong person. lim ju boo Founder, and First President
The Astronomical Society of Malaysia