I'm with Joseph, weeding, absolutely preventing seed formation for star thistle. Are your neighbors going to provide continual supplies of seeds? That can be difficult. Plant a buffer zone of some tall thicket.
Warm, disturbed soil is what star thistle needs for germination. If you can establish a continuous mat of vegetation on the soil in early spring while it is still cool, the star thistle won't have a chance to germinate. This will be a challenge without irrigation, for sure. But consider what grows early in the spring, the first things you see. Or I think buckwheat or a fall planted grass and pasture mix, so they can germinate when star thistle can't. Evne a mustard type plant is preferable to star thistle.
Are you aware what the seedling looks like? It bears no resemblance to the plant in flower. There are grey green broad leaves, kind of a furry surface, the edges of the leaves are jagged, maybe like a dandelion leaf when the plant is young. Anyway, they are not smooth shiny leaves. They grow flat and close to the ground at first. Lots of seedlings in dry situations have this adaptation. They grow flat against the ground and as wide as they can, it prevents something else from germinating right next to them, that might out compete them later. It also shades the soil and prevents moisture loss. I just googled for images of yellow star thistle seedling, and there are some good pictures there. Take a look if you don't already know what they look like.
So, start with eradication early in the season with the seedlings. Once you mow, then they will flower again, and will try to grow low against the ground again so that you can't mow them, and they'll flower down at the level below the mower blade.
If they do begin to flower, then they will right away begin to form seeds. Once the plant is cut, or pulled, there is usually enough moisture for the seeds to mature, so mowing is not enough, if they are in flower. You have to collect the plants with the flowers on them, and burn them, once burning season arrives again.
Hand weeding is the only way I have ever succeeded with eradicating star thistle.
I think they cannot regrow from
roots, so I used to cut the top of the plant off below the crown. Even if you miss a single cell of maristem and the plant grows back, it will have a long ways to go to get a chance at seeds again, but it is worth it to cut the plant below the crown if you can.
Also, don't get discouraged if you are sure no seed arrived from the neighbor and none were allowed to mature on your property, there are viable seeds in the soil for several years.
I don't envy you the star thistle
project.
I was a kid when star thistle arrived in the Los Altos Hills, (late 50s) and in my life the plant worked its way down to San Luis Obispo county where I was born and grew up. It is terrible stuff, ruins a field for hay, or grazing!
I had an acre pasture in Atascadero, in the early 2000s, with star thistle when I moved in and when I moved away I had eradicated it. I used mulch to prevent any growth in areas I could not get weeded. It was a real project. When I moved out, the next renter brought horses, and they beat the ground to death, and the star thistle came right back. The only thing that is left of what I did is a Matilija poppy there by el camino, getting bigger every year.
Thekla