I need a planetary system for my novel where an Earth-like warm habitable planet has an orbital period of about 650 to 730 days. I know that is similar to Mars and that Mars would be too cold. What type of system would work to achieve these effects?
Mars is too cold simply because its atmosphere is too thin to effectively trap enough heat. Otherwise it would still be within the habitable zone for the Sun, though close to its borders in some estimates.
Give it a stronger magnetic field to shield it from solar wind, and its atmosphere can last longer.
All you need is a slightly larger, heavier star that emits roughly (very roughly) four times the radiation the Sun does.
Very roughly because, in order to make a star hotter and brighter, you have to make it heavier, and that moves the orbit for a given period further from the star -- but the change due to orbital adjustment to keep the period around 700 (24 hour-ish) days is pretty small compared to the inverse square law effects on insolation.
Of course, the other side effect here is that a larger, hotter star will emit somewhat bluer light -- the surface temperature will be higher, there will be more ultraviolet, so either your planet needs a thicker ozone layer or everyone will need sunscreen, all the time.
Beyond that, larger, hotter stars don't last as long (they burn through their hydrogen fusion fuel faster, even in relation to the larger starting mass, than smaller stars). I don't know offhand what an F0 star will do in this regard compared to the G3 we orbit, or even what spectral type you'd wind up with to get about four times the Sun's luminosity -- but the lifetime will surely be shorter than the ten billion years (nearly half elapsed) expected for our Sun, which might affect whether the planet has had time to evolve advanced life.
If the star in the system is just like the Sun, a planet orbiting at the distance of Mars, with a semi major axis of 227,939,200 kilometers, would have an orbital period of 686.971 Earth days, which is within the specified 650 to 730 days and equal to 1.8809 Earth years.
The distance of Mars from the Sun is 1.523679 Astronomical Units or AU, and thus Mars receives an amount of radiation from the Sun equal to one divided by 1.523679 squared, or one divided by 2.3215976, or 0.4307378 of what Earth gets. Thus Mars would be expected to be colder than Earth.
That is made worse by the very thin atmosphere of Mars which retains heat less well than Earth's atmosphere. Humans can not survive in the Martian atmosphere anyway, so the planet will have to have an atmosphere much denser than that of Mars in order for humans or similar beings to survive, and such a much denser atmosphere will retain heat much better than that of Mars, helping to keep the planet warmer than Mars.
What is the minimum size of a planet necessary to produce and maintain a dense enough atmosphere to be habitable for humans and similar intelligent beings? A planet could lose it atmosphere but replace it fast enough to keep it dense, like Titan, the giant moon of Saturn does. But it should be assumed that a planet would have to have sufficient mass to produce an Earth like atmosphere and retain it for billions of years to be habitable for humans and aliens similar to them.
I think that it has been calculated that smaller planets with thinner atmospheres can be habitable for life closer to a star than larger planets with denser atmospheres, while larger planets with denser atmospheres can be habitable for life at greater distances from stars than smaller planets with thinner atmospheres.
Thus a planet near the size of Mars would be more likely to be habitable in an orbit near that of Venus, and a planet larger than Earth would be more likely to be habitable in an orbit near that of Mars.
One good source is Habitable Planets for Man, Stephen Dole, 1964.
Chapter 4, Astronomical Parameters, starts off with Planetary Properties on page 53. On page 54 it is calculated that the lowest escape velocity for a planet to retain atomic oxygen is 6.25 kilometers per second, corresponding to a planet with a mass of 0.194 Earth masses, a radius 0.63 of Earth, and a surface gravity of 0.49 g.
In the next few pages it is estimated that the minimum mass of a planet that would produce an atmosphere with enough oxygen would be somewhere between 0.25 and 0.57 Earth masses, and approximately 0.4 Earth masses, with a radius 0.78 of Earth and 0.68 Earth surface gravity, is selected as a rough minimum mass for a planet with enough oxygen in the atmosphere.
Since it was written in 1964 it has been discovered that the solar wind can knock particles out of the upper atmosphere of a planet. A strong planetary magnetosphere can reduce the solar wind and slow atmospheric loss and so it is also useful for a habitable planet.
Fortunately more massive planets more likely to produce and maintain a thick atmosphere are more likely to have strong magnetospheres.
On page 67 the discussion of the properties of the primary (the star) begins.
on page 68 it is said that "The only stars that conform with the requirement of stability for at least 3 billion years are main-sequence stars having a mass less than about 1.4 solar masses - spectral type F2 and smaller - ..."
The dimmer a star is, the closer a planet would have to orbit in order to receive enough heat from the star to have temperatures warm enough for liquid water. But the closer a planet is to its star, the stronger the tidal forces of the star upon the planet will be. If the tidal forces are strong enough they will slow the rotation of the planet until the planet is tidally locked, with one side always facing the star and the other side always in darkness. That is considered likely to make the planet uninhabitable.
On pages 71 to 72 it is said: "A "full" ecosphere can exist around primaries with a stellar mass greater than about 0.88 solar mass, but the ecosphere is narrowed by the tidal braking effect for primaries of lesser mass until it disappears when the stellar mass reaches about 0.76. The range in mass of stars which can have habitable planets is thous 0.72 to 1.43, corresponding to main-sequence stars of spectral types F2 through K1..."
So if your planet orbits around a spectral class F2 star, the star will be much more luminous than the sun and the star's "ecosphere" or habitable zone will be farther from the Star. Thus a habitable planet orbiting an F2 star would orbit much farther out and the total circumference of its orbit will be much greater than that of Earth. So the year of the planet will be longer that the year of Earth. However, because an F2 star will be more massive than the Sun, the orbital speed at the distance a habitable planet will orbit the F2 star will be greater than at the same distance from the Sun, and that will tend to reduce the length of the planet's year.
The distance at which a planet would receive the same amount of radiation from its star as Earth receives from the Sun can be calculated by multiplying Earth's distance from the Sun, one Astronomical Unit or AU, by the square root of the star's luminosity compared to the sun.
If the star is 2.00 times as luminous as the Sun, a planet 1.4142135 AU from that star will receive the same heat from the star as Earth receives at 1.0000 AU, if the star is 3.00 times as luminous as the Sun the planet should be 1.7320508 AU from the star, if the star is 4.00 times as luminous as the Sun, the planet should be 2.00 AU from the star, and so on.
Some studies show that there is a possibility that life could also develop on planets that orbit a F-type star. It is estimated that the habitable zone of a relatively hot F0 star would extend from about 2.0 AU to 3.7 AU and between 1.1 and 2.2 AU for a relatively cool F8 star. However, relative to a G-type star the main problems for a hypothetical lifeform in this particular scenario would be the more intense light and the shorter stellar lifespan of the home star.
I note that if two F2 stars orbited closely together, their combined habitable zone for planets orbiting both of them, in circumbinary or P-Type orbits, would be 1.4142135 times as wide as the habitable zone of just one F2 type star.
Thus the circumstellar habitable zone of a close binary FO star would extend to about 2.8 to 5.18 AU, and that of a close binary F8 star would extend to about 1.55 to 3.11 AU.
Here are links to discussions of the potential habitability of planets orbiting F class stars:
And of course the fictional planet could be farther from its star than the distance at which it would receive exactly the same amount of heat from the star as Earth gets from the Sun.
How much farther could it get? The obvious way to find out is to find the outer edge or limit of the Sun's circumstellar habitable zone and multiply it by the luminosity of the star relative to the Sun.
Take a look at this list of various estimates of the inner edge, the outer edge, or both, of the Sun's circumstellar habitability zone:
Note that they vary widely in their inner and outer edges and thus in how wide they are. The two most widely used estimates, by Hart et al in 1979, and by Kasting et al in 1993, are very different.
So it would be a good idea for science fiction writers to study the papers where those estimates are given and find which estimates seem most reasonable to them.
I note that Dole's estimation in 1964 was specifically for planets habitable for humans and Dole was not interested in planets with life if humans could not live on them. I think that many of the other estimates are for planets with life, regardless of whether humans could survive on them. It would be an error for a science fiction writer to use an estimated habitable zone outer limit for a planet where some life, but not humans, could survive, and put a planet habitable for humans or for lifeforms with requirement similar to humans at that distance from the star.
I hope this helps in a search for a plausible habitable planet with a year two Earth years long.
A Earth-like planet in a Mars orbit will get 43% of radiation compared with Earth. So, just make the star 2,32 brighter than the Sun.
Bubup, Belenos and Samaya shows fit enough. They are similar to the Sun, Bubup and Belenos are a bit older tho. All them still be in the main sequence, so allow conditions to one planet like you want come to existence.