This has all been very helpful, thank you. So with all this in mind had the PLO overthrown King Hussein in 1971 does that mean the "Lebanon war" would be the "Jordan war?" If this is the case Jordan has very different geography from Lebanon and I imagine that would drastically change how the war would play out.
An other question would be, How would Saudi Arabia respond to PLO dominated Jordan? Allies or not?
Also How does Iraq respond? The Soviet Union?
I would imagine that there may not end up being a Lebanon War that starts off as it did in OTL in 1982, yes.
There might still be a Lebanon War in the 1970s but only IF there is a Lebanese Civil War (not a guarantee), and IF this Civil War doesn't end very quickly (also not a guarantee with the PLO now having the resources of its own state, Jordan, at its disposal in prosecuting a war in Lebanon) and IF Israel decided that it wanted to intervene in this Lebanese Civil War in support of the Phalangists.
Assuming that there is either no Lebanese Civil War or one that ends quickly without Israel intervening, then there might indeed be some kind of "Jordan War" in the 1970s and 1980s.
And this would be fought very differently from the 1978 and 1982 Lebanese campaigns. In 1956 the Israelis had apparently used 175,000 troops in that Sinai campaign. In 1967 they had used approximately 70,000 soldiers in the Sinai, approximately 40,000 in the West Bank and approximately 30-40,000 in the Golan Heights campaigns.
The 1978 Lebanon campaign saw 25,000 troops used, while 1982 saw 78,000 troops used. In 1982, the peace treaty with Egypt had meant that Israel was able to significantly reduced military spending and was able to focus a lot more forces in the other border areas, which allowed them to use so many troops for Lebanon.
Assuming the 1971 POD does not significantly alter the Yom Kippur War, which seems unlikely (for instance a PLO run Jordan is likely to enter that war and there is no guarantee as to what would happen then - Egypt may have more success in the Sinai as a result; PLO Jordan might regain some land in the West Bank; Israel might well counterattack across the Jordan into the East Bank; maybe the superpowers get involved and the whole world is reduced to ash before 1974 even begins....), then IF Israel makes peace with Egypt, but not with Lebanon, Syria and PLO-Jordan then a Jordan War which is limited only to Jordan (so Syria and Israel end up fighting each other in Jordan rather than across the Golan Heights again) would see the Israelis tasked with possibly securing some kind of "security zone" all along the border with PLO-Jordan (that might well be the minimum objectives) or even occupying a major part of Jordan and attempting to expel the PLO and Palestinians from Jordan.
The geography of Jordan would be unlikely that of Lebanon where the Litani forms a natural objective (as do the smaller rivers running east to west further north along the coast) and the Lebanon and Anti-Lebanon mountains form natural barriers to the movement of forces (thus splitting Lebanon into two theatres - the coast to the Lebanon mountains and the Bekaa Valley between the Lebanon and Anti-Lebanon mountains and with the Anti-Lebanon Mountains forming a natural barrier with Syria, thus making it easier for Israel to campaign in Lebanon without having to send significant forces to control the border area with Syria but instead to focus on controlling access to the Damascus-Beirut Highway):
Instead, Jordan has basically has a major mountainous plateau which forms an escarpment along the edge of the Jordan Rift Valley (which forms Jordan's western border). The edge of the escarpment forms a series of heights which would likely constitute the minimum Israeli objective as it would give Israel control of the high ground overlooking the Jordan Valley much as control of the escarpment in the Golan Heights gave Israel control of the high ground overlooking Lake Tiberias and the upper Jordan Valley:
As can be seen from the map the chain of heights is interrupted by wadis and rivers running east to west. This makes it similar to
the Golan Heights where:
"The Golan Heights' unique terrain (mountainous slopes crossed by parallel streams every several kilometers running east to west), and the general lack of roads in the area channeled both forces along east-west axes of movement and restricted the ability of units to support those on either flank. Thus the Syrians could move north-south on the plateau itself, and the Israelis could move north-south at the base of the Golan escarpment. "
The Israelis would be looking at basically re-fighting the 1967 Golan campaign all along their eastern border and thus see fighting divided into maybe four sectors of east-west axes, while their north-south movement would be much free along the edge of the escarpment (and the PLO north-south movement would be freer on the plateau). If the Israelis only aim to take the heights along the escarpment they will control the entire Jordan Rift Valley which contains a major north-south road and will overlook most of Jordan's population centres (including Amman) in the next minor valley:
See map
here (originally from
here)
If the Israelis decide to go further their next objective line might be the highways and railways connecting Aqaba-Ma'an-Al Hasa-Al Qatranah-Al Jiza-Amman-Ar Rusayfah-Az Zarqa-Al Mafraq (if I'm not mistaken this line was speculated as the line that Israel would have have been interested in controlling and defending in 1991 had the Gulf War gone a little crazy with an Iraqi invasion of Jordan).
In 1967, Moshe Dayan opposed assaulting the Golan Heights as he estimated that such an assault might lead to 30,000 casualties. This of course did not happen, but given the 1967 Golan experience and presumably the 1973 Golan experience (and possibly a 1973 Jordan experience) then the Syrians may well be better prepared for an Israeli attempt to conquer the heights along the eastern edge of the Rift Valley in Jordan and be better able to coordinate the defence of this area.
At a minimum, any Jordan War scenario will probably see the Israelis needing to commit 40,000 soldiers and possibly more like 70-120,000 (maybe even 150,000 if they aim to take most of Jordan's populated areas). The amount of forces the Israelis can spare will depend a lot on what is happening in the Sinai with Egypt (whether there is peace with Egypt and thus no need to commit large numbers of forces to defend against a possible Egyptian attack or if there is no peace and substantial forces deployed along the western border).