No, not really.
First of all, the only entity (agency, firm, or organization) that genuinely has both the dedication to do it and the potential to carry out the task is SpaceX (Elon Musk).
Two SpaceX rocket boosters land in synchronized fashion after a Falcon Heavy launch in February.
Once it became clear that SpaceX was not going bankrupt any time soon, and kept talking about building a spacecraft system to take cargo and persons to Mars and back, Boeing jumped in and said “we’ll beat you there!” (What they actually said was something like “the first astronaut on Mars will get there on a Boeing rocket.)
Under Obama, NASA had a detailed human-exploration-of-Mars plan, which included the Boeing-led rocket design—that’s the only reason Boeing would attempt a super-heavy-lift booster, if NASA gave them a cost-plus contract to do it. But Trump immediately canceled that plan (can’t admit any goal of the Obama administration might have been good, after all) so the Boeing booster—creatively re-named the Space Launch System, or “SLS”—is still moving slowly forward. Just not pointed at Mars.
Other space agencies have Mars mission plans too, most notably the CNSA (China’s space agency). But they don’t seem to have much Mars-mission traction right now. They are focusing on small space stations and improving their Earth-to-LEO and Earth-to-Moon human spaceflight; once they have got that down pat, I have no doubt they will be ready to join a “race to Mars” if there is one …in the 2040s or later.
A number of NGOs are pushing for humans to go to Mars someday, too. Mars One gets a lot of press, but many doubt their dedication to the goal, and it doesn’t really have any traction or potential traction anyway.
The Mars Society is the most impressive of the NGOs pushing toward Mars. It is certainly dedicated and chock full of brilliant scientists, engineers, and grad students and post-docs doing real research, but the Society itself is not funded well enough to actually build the major hardware.
So, really, the “race to Mars” is Elon Musk vs a “low-probability but extreme-consequence” extinction event. Boeing and Lockheed could compete if they cared to do so, although I think shareholders would freak out at the prospect and mutiny. The Mars Society would compete if it were able to do so, but it simply doesn’t have the financial backing or really the organization to do so: it’s an advocacy and education group that also does creative research & development, not an aerospace firm or a space agency. The ESA, JAXA, NASA, and Roscosmos might compete if a compelling political imperative arose.
But right now, it’s really just lonely Elon and the hardworking tribe of engineers at SpaceX who are actually, slowly but surely, making progress toward Mars.
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