## Electromagnetic Propagation to Mars

April 5th, 2014

I do quite a bit of math for engineering projects.  Many things need iterations and so I do lots of scripting.  But sometimes I just want to do something for fun.  Awhile back I was watching a special about Mars and they were talking about the NASA‘s Spirit and Opportunity rovers that were supposed to operate for only about three months but have been operating for about 10 years now (Spirit has not responded for about four years now, though).  Some of the reasons they lasted so long is that it had been assumed that so much dust would accumulate on the horizontal solar cells making them inoperative over time.  What they had not anticipated is that, occasionally, there would be Dust Cleaning Events.  This could be just wind but a presenter from NASA at Fermilab once showed a Martian dust devil (remember the atmosphere there is about 95% Carbon Dioxide) which does a much better job.
The show that I was watching about Mars talked about the transit time for radio signals to Mars.  To cause something to happen with regards to the rovers, you have to send a signal and it takes a certain amount of time for the signal to get to Mars or conversely for a rover or an orbiter to communicate with Earth.  We send/receive signals to/from Mars usign the Deep Space Network (DSN).  Through these large satellite dishes, we can communicate with either the rovers or through orbiters which then relay the message to the rover(s).  For the Moon, we put in a NASA SBIR proposal for an easily deployable antenna array to communicate with orbiters.  So how long does it take.  I have heard the number of 15 minutes one-way thrown out at one time.  Here are my calculations:

Constants

d_earth_perihelion = 91.4 million km
d_earth_aphelion = 94.5 million km
d_mars_perihelion = 206,655,215 km
d_mars_aphelion = 249,232,432 km

Semi-major axis
Mars Average = 228 million km
Earth Average = 92.96 million km

Minimum  Distance

The closest that Earth and Mars will ever be is when Earth is at aphelion (farthest from the sun) and Mars is at perihelion (closest to the sun).  Also, they would have to be on the same radial from the sun at that time.  The chance of this happening is very unlikely.

Maximum Distance

Assuming the Earth and Mars are in opposition (across the Sun from each other).  Then both need to be at aphelion.

Two sources used:  One was a webpage specifically talking about the topic and the other came from my calculations based on “various sources” such as planetary tables.  One should always look at multiple sources in research.

The answers are based simply on the speed of light in a vacuum and takes each value and divides by that to determine the propagation time for the Electromagnetic (EM) Wave (debate wave verus particle another time) that is the signal sent to the rovers.  It is a simple calculation not taking into account any delay if the signal is relayed by an orbiter nor any compensation since space is not entirely a vacuum or the propagation through the Earth and/or Martian atmosphere (since that all should be negligible).

One way time for closest   3.033333 minutes
One way time for farthest  22.277778 minutes
One way time for average   12.500000 minutes

One way time for closest   6.233333 minutes
One way time for farthest  19.094444 minutes
One way time for average   7.502222 minutes

Constants – Various Sources
Closest – Source 1         54.600000 million km
Closest – Various          112.200000 million km
Farthest – Source 1        401.000000 million km
Farthest – Various         343.700000 million km
Average – Source 1         225.000000 million km
Farthest – Various         135.040000 million km

So, fifteen minute one way was probably right at that time.   Comments/corrections welcome.

Ronald Kollman – RF Hardware Designer and President of Haynes-Bent, Inc.

Haynes-Bent, Inc. – Radiofrequency (RF) hardware design, EMC/EMI analysis and 3D Electromagnetic simulation. We also provide Technology advise for investors and executives and software services from embedded to network servers.

sales@haynesbent.com (630)845-3316